February 24, 2021w3.impa.br/~landim/Cursos/AF.pdfin any standard textbook on metric spaces. In portuguese, I recommend the proof of [Oliveira, Teorema 2.5]. Further readings. A.[Zeidler-108,

Lectures on Functional Analysis

General recommendations.

• These lectures assume that the audience is familiar with measure theory.• The videos do not replace the books. I suggest to choose one among the

many listed at the end of these notes and to read the corresponding sectionsbefore or after the videos.• After the statement of a result, interrupt the video and try to prove the

assertion. It is the only way to understand the difficulty of the problem, todifferentiate simple steps from crucial ones, and to appreciate the ingenuityof the solution. Sometimes you find an alternative proof of the result.• You can speed-up or slow-down the video. By pressing settings at the

bottom-right corner, you can modify the playback speed.• Send me an e-mail if you find a mistake which is not reported in these notes.• If you typed in latex, with no personal definitions nor the use of special

packages, solutions to some exercises proposed below, send the file. Hope-fully, I’ll create a note with solutions to the exercises, acknowledging theauthors of the solutions.• A note about the methodology. I ask the students to view the video(s)

before the class. In the first part of the lecture, I recall the content of thevideo. Sometimes, I ask one of the students to replace me. Occasionally,the student is randomly chosen. This is the opportunity for the students toask questions on the content of the class. In the second part of the lecture, Ipresent some of the applications included in the “Further Readings” topic.

February 24, 20211

2

Lecture 1: Linear spaces: definition, examples and linear span

Summary. This lecture is based on [Lax, Chapter 1].

Content and Comments.

0:00 Definition of linear spaces.5:25 Elementary properties: 0 ∗ x = 0, −x = (−1) ∗ x8:54 Examples of linear spaces.

13:24 Sublinear spaces, examples and properties (0 is linear subspace. The sumof two linear subspaces is a linear subspace. Intersection of linear subspacesis a linear subspace). [Lax, Theorem 1.1]

22:25 Linear subspaces ordered by inclusion. The union of a collection of linearsubspaces totally ordered by inclusion is a linear subspace. [Lax, Theorem1.1.(iv)]

26:15 Linear spans. Characterization of the linear span. [Lax, Theorem 1.2].30:02 Linear combinations. Description of the linear span of a set S. [Lax,

Theorem 1.2(ii)].

Further readings.

A. [Lax, Chapter 1] gives many examples of linear spaces not presented here.B. [Zeidler-108, Section 1.1] defines the dimension of a linear space.C. [Bachman-Narici, Section 1.1] defines independence of vectors and Hammel

bases of linear spaces.

Recommended exercises.

a. Show that the examples 1–11 of [Lax, Chapter 1] are linear spaces.

3

Lecture 2: Linear spaces: quotient spaces and convex sets



0:00 Definition of a quotient space. The quotient space is a linear space.15:15 Definition of linear maps and isomorphism.18:01 Linear maps and linear subspaces. T (Y ) is a linear subspace if Y is one.

[Lax, Theorem 1.3.(i)]21:33 T−1(Y ) is a linear subspace if Y is one. [Lax, Theorem 1.3. (ii)]24:47 Convex sets.27:08 A convex combinations of elements of a convex set belongs to the convex

set. [Lax, Theorem 1.4]31:56 Properties of convex sets. Linear subspaces are convex. [Lax, Theorem

1.5.(iii)]33:05 The sum of convex sets are convex. [Lax, Theorem 1.5.(iv)]36:49 Intersections of convex sets are convex. [Lax, Theorem 1.5. (vi)]38:53 The union of convex sets totally ordered by inclusion is convex. [Lax,

Theorem 1.5. (vii)]41:45 The image by a linear map of a convex set is convex. [Lax, Theorem 1.5.

(viii)]44:22 The inverse image by a linear map of a convex set is convex. [Lax, Theorem

1.5. (ix)]46:28 Definition of the convex hull of a set.48:00 The convex hull of a set S is the smallest convex set containing S. [Lax,

Theorem 1.6.(i)]50:57 The convex hull of a set S consists of all convex combinations of points of

S. [Lax, Theorem 1.6.(ii)]

Further readings.

A. [Lax, Chapter 1] defines the extreme points and extreme sets of a convexsets. Some results concerning these sets are given, as well as many examples.

B. [Bachman-Narici, Section 8.8] proves that the closure of a convex set isconvex and defines closed convex hulls.

C. [Zeidler-108, Section 1.10] gives examples of convex sets and defines convexfunctions.


a. Prove Theorem 5.(i), (ii) and (v) in [Lax, Chapter 1].b. [Zeidler-109, Chapter 1], exercise 1.

Additional exercises.

a. Prove all claims in [Lax, Chapter 1] concerning extremes points and extremesets.

4

Lecture 3: Normed linear spaces: definition and basic properties

Summary. This lecture is based on [Lax, Section 5.1].


0:00 Definition of a norm and normed linear spaces2:19

∣∣ ‖x‖ − ‖y‖ ∣∣ ≤ ‖x − y‖ and ‖ − x‖ = ‖x‖.5:06 Definition of the distance d(x, y) = ‖x − y‖.8:52 The topology induced by the norm. Equivalent norms.

11:31 Subspaces of a normed linear spaces are normed linear spaces. Product ofnormed linear spaces are normed linear spaces.

16:04 The closure of a normed linear space is a normed linear space. [Lax, The-orem 5.2].

17:38 The quotient space of a NLS with respect to a closed NLS is a NLS. [Lax,Theorem 5.1].

40:40 Definition of Banach spaces

Further readings.

A. [Zeidler-108], sections 1.1 to 1.4. Many examples and properties of norms.B. [Bachman-Narici], sections 3.1 and 3.2. Other examples of normed linear

spaces.


a. [Lax, Chapter 5], exercises 1, 3.b. [Zeidler-108, Chapter 1], exercises 1a to 1i. Note that examples 1c and 1d

are not dense.c. [Zeidler-108, Chapter 1], exercise 5.d. [Bachman-Narici, Chapter 8], exercises 3, 19.


a. [Zeidler-108, Chapter 1], exercises 2, 9.d. [Bachman-Narici, Chapter 8], exercises 4, 13, 20.

5

Lecture 4: Completing a normed linear space

Summary. This lecture presents the standard method of completing a normedlinear space. The construction does not differ from the one employed to complete ametric space and is inspired from the construction of the real numbers by Cantor.A proof of these results can be found in [Bachman-Narici, Section 4.2 and 8.3] orin any standard textbook on metric spaces. In portuguese, I recommend the proofof [Oliveira, Teorema 2.5].

Further readings.

A. [Zeidler-108, Section 1.3] for elementary examples of complete normed linearspaces.

B. [Bachman-Narici, Section 4.2] presents examples of normed linear spaceswhich are not complete.


a. [Bachman-Narici, Chapter 4], exercise 4b. [Zeidler-109, Chapter 1], exercise 2.


a. [Bachman-Narici, Chapter 4], exercises 1, 2, 6, 9, 10.

6

Lecture 5: Finite dimensional linear spaces

Summary. This lecture is based on [Zeidler-108, Sections 1.1 and 1.12] and[Bachman-Narici, Section 8.5].


0:00 Definition of linear independence. Uniqueness of representation. This isDefinition 6 of [Zeidler-108, Chapter 1].

3:39 Maximal set of linearly independent elements. Definition of dimension of alinear subspace and proof that it is well defined.

12:17 For a proof that N = M , see below.15:44 Let z1, . . . , zN be linearly independent. There exists C0 < ∞ such that

for all x ∈ span z1, . . . , zN, x =∑

1≤j≤N αj(x) zj ,

N∑j=1

|αj(x) | ≤ C0 ‖x‖ .

26:57 In a finite-dimensional linear space the ball B(0, 1) is compact. [Zeidler-108,Corollary 1.8]

31:40 In finite-dimensional linear spaces, all norms are equivalent. [Bachman-Narici,Theorem 8.7] and [Zeidler-108, Proposition 1.4]

40:53 All finite-dimensional linear spaces are complete. [Zeidler-108, Corollary1.6]

Let A, B be matrices M ×N and N ×M respectively, and assume, without lossof generality, that N ≥M . Then, N = M if AB = Id.

Note that we can always assume that N ≥M . If this is not the case, repeat theargument to obtain, at the end, the identity BA = Id instead of AB = Id.

Denote by x1, . . . , xN the lines of the matrix A and by y1, . . . , yN the columnsof the matrix B. These are vectors in RM . Since AB = Id, 〈yj , xk〉 = δj,k, where〈u, v〉 stands for the usual scalar product between vectors u, v in RM . We claimthat y1, . . . , yN are linearly independent. Indeed, assume that

N∑j=1

αj yj = 0 .

Take the scalar product with xk on both sides to conclude that αk = 0.Since RM has dimension M , N ≤ M . As, by assumption, N ≥ M , we obtain

that N = M .In this argument, we used the fact that RM has dimension M . We leave to the

reader to prove this result or to find a reference.

Further readings.

A. [Zeidler-108, Section 1.12] defines a basis of a finite-dimensional space andproves its existence. A finite-dimensional linear subspace of a linear spaceis closed.


a. [Zeidler-108, Chapter 1], exercise 11.

7

b. [Bachman-Narici, Chapter 8], exercises 11 [requires the Baire category theo-rem which will be presented later in these lectures. It’s however worthwhileto learn at this point that infinite-dimensional Banach spaces do not havecountable bases!], 14, 15, 18.


a. [Bachman-Narici, Chapter 8], exercises 12, 16.

8

Lecture 6: Examples of normed linear spaces

Summary. This lecture is based on [Lax, Chapter 5] and [Bachman-Narici, Sec-tions 8.1 and 8.2].


0:00 A Cauchy sequence in a NLS is uniformly bounded.3:20 Example 1: `∞ is a Banach space.

16:32 Example 2: `p, 1 ≤ p < ∞, is a Banach space. The proof of Holder’sinequality and the triangle inequality can be found in [Taylor, Theorems7.7 and 7.8] or in [Bachman-Narici, Theorems 8.2 and 8.3] for the finite-dimensional case.

35:22 Example 3: Let (S, d) be a metric space. The space of continuous func-tions f : S → R with compact support, endowed with the norm ‖f‖∞ =supx∈S |f(x)|, is a normed linear space which is not complete if S is notcompact. It is a Banach space if S is compact.

38:52 Example 4: Let D be a domain of RN and C0(D) be the space of continuousfunctions f : D → R with compact support. Fix 1 ≤ p < ∞, and definethe norm ‖f‖p on C0(D) by ‖f |pp =

∫D|f(x)|p dx. (C0(D), ‖ · ‖p) is a

normed linear space which is not complete. Its completion is denoted byLp(D). Domain means an open, connected set.

Further readings.

A. [Lax, Chapter 5] provides many examples of Banach spaces. Essentially, allwhich will appear in these lectures.

B. [Bachman-Narici, Sections 8.1 and 8.2] proves that C[a, b] with the supnorm is complete. It gives a different definition for Lp(D) and examinesthe space of bounded variation functions defined on an interval and thespace of signed measures.


a. [Lax, Chapter 5]. Prove Theorem 4.b. [Lax, Chapter 5]. Prove the claims of examples (a)–(g).c. [Bachman-Narici, Chapter 8], exercise 10.


a. [Zeidler-108, Chapter 1], exercises 6, 7, 8.b. [Bachman-Narici, Chapter 8], exercises 8, 9.

9

Lecture 7: In infinite dimensions the unit ball is not compact



0:00 Riesz lemma: Let Y be a closed, proper, linear subspace of X, Y 6= X,Y 6= ∅. Then, ∀ ε > 0, ∃ z ∈ X such that ‖z‖ = 1 and ‖z − y‖ ≥ 1− ε forall y ∈ Y . [Lax, Lemma 5.7]

14:36 In infinite dimensions the unit ball is not compact. [Lax, Theorem 5.6]30:31 Separable spaces. An example of NLS which not separable: the signed

measures on [−1, 1]. The definition of a signed measure and the Hahn-Jordan decomposition theorem can be found in [Taylor, Sections 3.1 and3.2]. The Banach space of signed measures is examined in [Bachman-Narici,Examples 8.9]

Further readings.

A. [Lax, Section 5.2] defines strictly subadditive norms and uniformly convexBanach spaces.

B. [Bachman-Narici, Section 6.4] gives some examples of separable metricspaces.


a. [Lax, Chapter 5], exercise 4.b. [Bachman-Narici, Chapter 6], exercise 6.c. [Bachman-Narici, Chapter 8], exercise 17.

10

Lecture 8: Zorn’s lemma

Summary. This lecture is based on the appendix of [Bachman-Narici, Chapter9] and on [Oliveira, Chapter 10]. There is a flaw in Appendix C of [Lax]. Thedefinition of maximal element is not correct. The definition is as follows. “Anelement m of a partially ordered set Ω is called maximal if m = x for all x ∈ Ωsuch that m ≤ x.”


0:00 Definition of partial and total order.4:34 An example of partial order in R2.6:25 Definition of an upper bound of a subset of a partially ordered set.7:55 Definition of a maximal elements of a set.

10:29 Statement of Zorn’s lemma.12:52 Totally ordered subsets of a partially ordered set.16:22 Definition of a Hamel basis of a linear space. All linear spaces have a Hamel

basis. [Oliveira, Proposition 10.9]

Further readings.

A. [Dudley, Chapter 1] presents an introduction to set theory which includesthe proof of the equivalence between the axiom of choice and Zorn’s lemma.


a. [Bachman-Narici, Chapter 9], exercises 1, 2, 20.

11

Lecture 9: The Hahn-Banach theorem



0:00 Linear functionals, the Hahn-Banach theorem. [Lax, Theorem 3.1].0:00 Note that the functional p : X → R is not required to be positive. An

example where p takes negative values is given in Lecture 11.0:00 Positive homogeinity ensures that p(ax) = ap(x) for positive scalars (a >

0). We can extend this identity to a ≥ 0:

p(ax) = ap(x) for all a ≥ 0 , x ∈ X . (0.1)

To prove this identity for a = 0, we just need to show that p(0) = 0 because0 · x = 0 and 0 · p(x) = 0. By positive homogeneity,

p(0) = p(2 · 0) = 2 p(0) =⇒ p(0) = 0 .

7:31 Part 1 of the Proof. Let ` : Y → R be a linear functional defined on asublinear space Y 6= X and satisfying the hypotheses of the theorem. Fixz 6∈ Y . We extend the linear functional from Y to the space span (Y, z).

25:46 Part 2 of the Proof. Use Zorn’s lemma to extend ` to X.

Further readings.

A. [Zeidler-109, Section 1.1] presents some applications of the Hahn-Banachtheorem which will be seen later.

B. [Lax, Section 3.3] presents two extensions of the Hahn-Banach theorem,including a version of the theorem for Banach spaces over the complexnumbers.

C. [Bachman-Narici, Section 11.1] also provides a version of the Hahn-Banachtheorem for Banach spaces over the complex numbers.

D. [Bachman-Narici, Appendix of Chapter 11] presents a proof of the existenceof an additive (not σ-additive) measure on R defined over all subsets of R,based on the Hahn-Banach theorem.

E. Applications of the Hahn-Banach theorem are given in Lectures 11 and 14Ato 14D.


a. [Zeidler-109, Section 1.1], exercise 5.b. [Bachman-Narici, Chapter 11], exercise 10.


b. [Bachman-Narici, Chapter 11], exercises 1, 3.

12

Lecture 10: Convex sets and gauge functions



0:00 Interior points of a subset of a LS.3:32 Definition of the Gauge function associated to a convex set and of an interior

point.10:40 The Gauge function is positive-homogeneous and subadditive. [Lax, Theo-

rem 3.2].23:21 For any convex set K and interior point x of K, p(y) ≤ 1 if x + y ∈ K.

Moreover, x + y is an interior point of K if and only if p(y) < 1. [Lax,Theorem 3.3].

36:59 Let p be a positive-homogeneous, subadditive function. Then, the set x :p(x) ≤ 1 is convex. Furthermore, the set x : p(x) < 1 is convex and theorigin is an interior point of this set. [Lax, Theorem 3.4].

42:34 I’m using here equation (0.1).44:34 We do not need the hypothesis that p takes only non-negative values. Here

is the proof in the general case p : X → R. We have to show that for allx ∈ X there exists ε > 0 such that tx ∈ B for all |t| < ε.

Fix x ∈ X. We divide the proof in four cases.Case 1. Assume that p(x) > 0 and p(−x) > 0, and let ε = minp(x)−1, p(−x)−1.

(a) For 0 < t < ε, by positive homogeneity, p(tx) = tp(x) < 1.Hence tx ∈ B for all 0 < t < ε.

(b) For t = 0, tx = 0 and we have already showed that 0 ∈ B.(c) For −ε < t < 0, by positive homogeneity, p(tx) = p(−t · −x) =

(−t)p(−x) < 1. Thus, tx ∈ B for all −ε < t < 0.In conclusion, in Case 1, tx ∈ B for all |t| < ε.

Case 2. Assume that p(x) > 0 and p(−x) ≤ 0, and let ε = p(x)−1.(a) For 0 < t < ε, as in Case 1a, tx ∈ B for all 0 < t < ε.(b) For t ≤ 0, by the positive homogeneity stated in equation (0.1)

and since p(−x) ≤ 0, p(tx) = p(−t · −x) = (−t)p(−x) ≤ 0 < 1.Thus, tx ∈ B for t ≤ 0.

In conclusion, in Case 2, tx ∈ B for all t < ε.Case 3. Assume that p(x) ≤ 0 and p(−x) > 0, and let ε = p(−x)−1. As in

Case 2, tx ∈ B for all t < ε.Case 4. Assume that p(x) ≤ 0 and p(−x) ≤ 0. As in Case 2, tx ∈ B for all

t ∈ R..


a. [Zeidler-109, Chapter 1], exercise 17.b. [Bachman-Narici, Chapter 11], exercise 7.


a. [Zeidler-109, Chapter 1], exercises 18 and 19.

13

Lecture 11: Geometric Hanh-Banach theorems

Summary. This lecture is based on [Lax, Sections 3.2 and 4.1].


0:00 Definition of hyperplanes. The hyperplane speration theorem. [Lax, Theo-rem 3.5].

15:52 Let K be convex set with at least one interior point. Then, for all y 6∈ K,there exists a linear functional ` such that `(x) ≤ `(y) for all x ∈ K. [Lax,Corollary of Theorem 3.5].

19:51 The extended hyperplane separation theorem. [Lax, Theorem 3.6].34:00 Application 1: Extension of a bounded linear functional defined on a linear

subspace. [Lax, Theorem 8.4] and [Zeidler-109, Theorem 1.1B].38:07 Application 2: Extension of a positive linear functional. [Lax, Theorem

4.1]. This is an example of a subadditive, positive homogeneous functionwhich takes negative values.


a. Prove Standard Example 1 of [Zeidler-109, Section 1.1] with K = R.b. Prove Proposition 3 of [Zeidler-109, Section 1.2] with K = R.c. [Zeidler-109, Chapter 1], exercise 6.d. Prove [Bachman-Narici, Theorems 11.3 and 11.4].

14

Lecture 12: Dual of a normed linear space

Summary. This lecture is based on [Lax, Section 8.1]. In this and the nextlectures, the dual of a normed linear space X is sometimes represented by X ′, asin Lax, and sometimes represented by X∗, as in Zeidler.


0:00 Definition of continuous linear functionals and of bounded linear function-als. A linear functional is continuous if and only if it is bounded [Lax,Theorem 8.1]

8:25 Definition of the dual of a normed linear space.12:10 The null space of a bounded linear functional is a closed linear subspace.

[Lax, Theorem 8.2].23:29 The dual X ′ of a NLS is complete. [Lax, Theorem 8.3]. Note that we do

not require X to be complete.


a. Prove Proposition 4 [Zeidler-108, Section 1.11].b. [Zeidler-108, Chapter 1], exercises 1k, 10.c. Prove [Bachman-Narici, Theorem 11.5].d. [Bachman-Narici, Chapter 11], exercises 4, 5, 8, 9, 11.d. [Bachman-Narici, Chapter 13], exercises 3, 4, 8.


a. [Bachman-Narici, Chapter 11], exercises 6.

15

Lecture 13: Extension of bounded linear functionals, closed linear spans



0:00 Let x1, . . . , xN be a family of linear independent elements of X andα1, . . . , αN real numbers. Then, there exists a bounded linear functional` : X → R such that `(xj) = αj , 1 ≤ j ≤ N . [Lax, Theorem 8.5].

11:24 Let Y be a finite-dimensional space. The complement of Y is closed. [Lax,Corollary 8.4’].

23:04 ‖x‖ = sup`(x) : ‖`‖ = 1

[Lax, Theorem 8.6 and Corollary 8.5’]. Note

that we actually constructed a bounded linear functional ` : X → R such`(x) = ‖x‖. In particular, the supremum is a maximum:

‖x‖ = max`(x) : ‖`‖ = 1

.

The fact that it is a maximum (that there exists a bounded linear functionalsuch that `(x) = ‖x‖) is used in the following lectures.

34:00 inf‖x − y‖ : y ∈ Y

= sup

`(x) : ‖`‖ = 1 , `(y) = 0 ∀ y ∈ Y

.

[Lax, Theorem 8.7]. Here also we construct in the proof a bounded linearfunctional ` : X → R which satisfies the identity. We may, therefore,replace the supremum by a maximum:

inf‖x− y‖ : y ∈ Y

= max

`(x) : ‖`‖ = 1 , `(y) = 0 ∀ y ∈ Y

49:24 Definition of Y ⊥ and ‖`‖Y = sup | `(y) | : y ∈ Y , ‖y‖ = 1 . .51:45 ‖`‖Y = inf

‖m− `‖ : m ∈ Y ⊥

. [Lax, Theorem 8.7’]. In the proof of this

result, we constructed a bounded linear functional m ∈ Y ⊥ which satisfiesthe identity. We may, therefore, replace the infimum by a minimum:

‖`‖Y = min‖m− `‖ : m ∈ Y ⊥

.

1:04:32 Definition of the closed linear span. Description of the closed linear span.[Lax, Exercise 8.4]

1:13:36 A criterion for a point to belong to the closed linear span of a subset. [Lax,Theorem 8.8].

Further readings.

A. Most of the results of this lecture are covered in [Bachman-Narici, Section12.1].


a. [Lax, Chapter 8], exercises 1, 2, 3.b. Fix 1 < p < ∞. Denote by `p the space of sequences of real numbers

x = (x1, x2, . . . ) such that∑j≥1 |xj |p <∞, endowed with the norm ‖x‖p =

(∑j≥1 |xj |p)1/p. Prove that the dual of `p is `q, where p−1 + q−1 = 1. See

[Bachman-Narici, Section 12.3].c. [Bachman-Narici, Chapter 12], exercises 4, 5, 6, 7.

16


a. [Bachman-Narici, Chapter 12], exercises 1.

17

Lecture 14: Reflexive spaces



0:00 Definition of the second dual. Definition of the map L : X → X ′′ givenby L(x) = Lx, where Lx(`) = `(x) for all x ∈ X, ` ∈ X ′. Proof that‖Lx‖ = ‖x‖.

8:27 Definition of reflexive spaces. Examples of reflexive spaces: finite-dimensionalspaces and Lp(Ω), 1 < p <∞. [Lax, Theorem 8.11].

12:10 The space C([−1, 1]) is not reflexive. [Lax, Theorem 8.12].25:20 Let X be a normed linear space over the real numbers. If X ′ is separable,

then so is X. [Lax, Theorem 8.13]. As many results in these lectures, theresult holds if the scalar field is C. We stated it with R as scalar fieldbecause we use the Hahn-Banach theorem in its proof, which has beenstated with R as scalar field.

38:14 Another proof that the space C([−1, 1]) is not reflexive. This one is basedon the previous result on separability.

43:40 Let X be a reflexive normed linear space over the real numbers and Y aclosed linear subspace of X. Then, Y is reflexive. [Lax, Theorem 8.15].

Further readings.

A. [Zeidler-109, Section 2.8]. The map L : X → X ′′ is represented by j inthis reference. Zeidler proves that the map j is linear. Hilbert spaces arereflexive. This will be seen later in these lectures.

B. [Bachman-Narici, Sections 12.2, 12.3]. The map L : X → X ′′ is representedby J . Incomplete normed spaces are not reflexive.


a. Show that a normed finite-dimensional space is reflexive.b. Fix 1 < p <∞. Show that `p is reflexive.c. Let X be a separable and reflexive Banach space over R. Show that X ′ is

separable.d. [Bachman-Narici, Chapter 13], exercise 7e. [Zeidler-109, Chapter 3], problems 11, 18.f. [Reed-Simon, Chapter 3], problem 22.


a. [Zeidler-109, Chapter 3], exercises 9, 12, 13.b. [Bachman-Narici, Chapter 12], exercise 10.

18

Lecture 14A: The dual space of C([a, b])

Summary. This lecture is based on [Zeidler-109, Section 1.3]. It presents aninteresting application of the Hahn-Banach theorem which will not be used in thecoming lectures, and can be skipped. Many gaps left in the lecture are filled below.

Content, Comments and References.

0:00 The space C(X) of continuous functions over a compact metric space en-dowed with the sup norm as a Banach space. Definition of signed measureson a metric space X and their basic properties.

Later, we need X to be compact. Hence, though this hypothesis is notneeded at this stage, to avoid confusion, assume throughout this lecturethat X is compact.

The definition of a signed measure and the Hahn-Jordan decompositiontheorem can be found in [Taylor, Sections 3.1 and 3.2]. The Banach spaceof signed measures is examined in [Bachman-Narici, Examples 8.9]

6:30 The space M(X) of finite signed measures.The proof that ν(A) > −∞ for all A in the σ-algebra if there exists B

such that ν(B) = +∞ is part of [Taylor, Theorem 3.1].8:42 Riesz representation theorem. Let X be a compact, metric space and C(X)

the set of continuous functions f : X → R. Then, there is a one-to-onecorespondence between finite, signed, Baire measures ν on X and the dualspace to C(X) given by

`(f) =

∫X

f dν

Moreover, ‖`‖ = ‖ν‖.The assumption that X is separable, mentioned in the lecture, is not

needed.This result is a particular case of the corollary of Theorem 9.9 in [Taylor,

Section 9.5].We refer to [Royden, Section 14.1] for the definition of the Baire sets. In

the case of compact metric spaces, the Borel σ-algebra coincides with theBaire σ-algebra.

12:54 The set of bounded variation functions.We refer to [Taylor, Section 9.1] for the definition of bounded variation

functions and a proof of the fact that they can be written as the differenceof two monotone functions.

15:46 The correspondance between bounded variation functions and finite signedmeasures. Note that ρ(a) = ν(a).

23:21 Proof of the Riesz representation theorem in the case where X = [a, b].1:00:40 Example of an element in C([a, b])′: `(f) =

∫[a,b]

h(t) f(t) dt, where h ∈L1([a, b]). Proof that ‖`‖ = ‖h‖1.

19

Further readings.

A. Sections 3.1 and 3.2 of [Taylor] for properties of signed measure and a proofof the Hahn-Jordan decomposition. Theorem 14.A is a particular case ofthe corollary of Theorem 9.9 in [Taylor, Section 9.5]. In Section 9.1 of[Taylor] on finds the definition of bounded variation functions and a proofof the fact that they can be written as the difference of monotone functions.

B. [Bachman-Narici, Chapter 13] covers in detail the content of this lectureand provides a table of spaces with their conjugate.

C. Section 14.1 of [Royden] for the definition of the Baire sets (which appearsin the statement of Theorem 9.9 in [Taylor, Section 9.5]). In the case ofcompact metric spaces, Borel σ-algebra coincides with the Baire σ-algebra.

D. A description of the bidual of C(X) can be found in S. Kaplan, On thesecond dual of the space of continuous functions, Trans. Amer. Math.Soc. vol. 86 (1957), 70–90 or in the book S. Kaplan, The bidual of C(X).I, North-Holland Mathematical Studies 101, North– Holland, Amsterdam,1985.


*a [Bachman-Narici, Chapter 13], exercises 2, 9, 10.


a. Let MX be the space of finite signed measures on a compact metric spaceX. Show that ‖ν‖ = ν+(X) + ν−(X) is a norm and that MX is completewith this norm.

b. Let ν be a finite signed measure on [a, b]. Show that the function ρ : [a, b]→R defined by ρ(t) = ν(a) + ν([a, t]) has bounded variation.

c. Show that the class of sets S, introduced at time 19:03, is a semi-algebra.Recall that S = (c, d] : a ≤ c < d ≤ b ∪ [a, d] : a ≤ d ≤ b .

d. Show that the function ν : S → R, introduced at time 19:51 and given byν(c, d] = ρ(d)− ρ(c), ν[a, d] = ρ(d), is σ-additive on S.

e. Prove that the class of Baire sets coincides with the Borel sets in a compactmetric space.

20

Lecture 14B: The moment problem and Chebyshev approximation.

Summary. This lecture is based on [Zeidler-109, Sections 1.4 and 1.6]. It can beskipped.


0:00 The finite moment problem. [Zeidler-109, Proposition 1.4.1]. See wikipediafor the definition of Lagrange polynomials.

22:09 The moment problem. [Zeidler-109, Proposition 1.4.2]32:32 Chebyshev approximation. [Zeidler-109, Proposition 1.6.1]35:27 Let ` ∈ C([a, b])′. Assume there exists u ∈ C([a, b]) such that `(u) = ‖`‖ ‖u‖

and |u(t)| achieves its maximum at N points. Then, ` =∑

1≤j≤N αj δtjand ‖`‖ =

∑1≤j≤N |αj |. [Zeidler-109, Lemma 1.5.3]

53:12 Proof of Chebyshev approximation.

Recommneded exercises.

a. Let Y be a finite-dimensional subspace of a NLS X, and x ∈ X. Showthat there exists y0 ∈ Y such that ‖y0 − x‖ = inf ‖y − x‖ : y ∈ Y . See[Zeidler-109, Corollary 1 of Theorem 1.D].

21

Lecture 14C: A dual variational problem in optimal control.

Summary. This lecture is based on [Zeidler-109, Section 1.7]. It can be skipped.


0:00 Minimizing the cost to send a rocket to an altitude h. [Zeidler-109, Propo-sition 1.7.1]


a. [Zeidler-108, Chapter 2], exercise 1.

22

Lecture 14D: Existence of a Green function.

Summary. This lecture is based on the proof of [Davis, Theorem 7.4] and on [Lax,Section 9.5]. It can be skipped.


0:00 Definition of an harmonic function in a domain of R2.2:35 Maximum principle for harmonic functions. [Gilbarg-Trudinger, Theorem

2.2].7.32 Definition of a Green function. [Davis, Section 3.7]

11.28 The existence of Green function provides a solution of the Laplace equation∆u = 0 on D and u = f on the boundary of D.

16:49 Proof of the existence of Green function. See P. D. Lax, On the existence ofGreen’s function. Proc. Amer. Math. Soc. 3 (1952), 526–531. A detailedproof can be found in [Davis, Section 3.7].

31:53 Here, I’m assuming that y 6∈ D.37:50 Keep in mind that x0 ∈ D.

23

Lecture 15: Hilbert spaces



0:00 Definition of a scalar product. A scalar product is also called an innerproduct. A linear space endowed with a scalar product is sometimes calleda pre-Hilbert space or an inner product space.

5:48 ‖x‖ :=√〈x, x〉 defines a norm.

9:45 Proof of Schwarz inequality. [Lax, Theorem 6.1].19:20 | < x, y > | = ‖x‖ ‖y‖ =⇒ y = αx. Note that if y = 0, then y = αx for

α = 0.25:34 ‖x‖ = max | 〈x, y〉 | : ‖y‖ = 1 . [Lax, Corollary of Theorem 6.1].28:48 Proof of the triangle inequality.32:00 Parallelogram identity.34:25 Definition of a Hilbert space.35:44 The scalar product is continuous with respect to the norm.41:18 The completion of an inner product space is a Hilbert space.58:29 Examples of inner product spaces: C([a, b]), `2, L2(D), where D is a domain

of Rd.

Further readings.

A. Sections 2.1 and 2.2 in [Zeidler-108].B. [Bachman-Narici, Section 9.4] gives an example of a nonseparable Hilbert

space.


a. [Lax, Section 6.1], exercises 1, 2, 3.c. Prove Propositions 4, 9, 10 of [Zeidler-108, Section 2.1].d. Prove Propositions 5, 7, Corollary 8 and Lemma 10 of [Zeidler-108, Section

2.2] and all claims stated in Examples 1, 2, 4 and 9.e. [Zeidler-108, Chapter 2], exercises 2, 3, 4, 5.


a. Prove Propositions 11, 12 of [Zeidler-108, Section 2.2].b. [Zeidler-108, Chapter 2], exercise 6.c. [Bachman-Narici, Chapter 12], exercises 2 and 3.

24

Lecture 16: Closed convex subsets of a Hilbert space.

Summary. This lecture is based on is based on [Lax, Section 6.2].


0:00 K closed convex subset of a HS X and x ∈ X. ∃ ! y0 ∈ K such that ‖y0 −x‖ = inf ‖y−x‖ : y ∈ K. [Lax, Theorem 6.2]. See also [Bachman-Narici,Theorem 10.5].

14:18 Definition of the orthogonal Y ⊥ of a linear subspace of a Hilbert space.Assume that Y is closed. Then Y ⊥ is a closed linear subspace and X =Y ⊕ Y ⊥. Moreover (Y ⊥)⊥ = Y . [Lax, Theorem 6.3]. Two elements x, y ofan inner product space are said to be orthogonal if 〈x , y〉 = 0.

30:27 Remark. In contrast with Hilbert spaces, in Banach spaces Y CLS 6=⇒ Y ⊥

CLS.


*a. [Bachman-Narici, Chapter 9], exercises 10, 11, 15, 17, 19.*b. [Bachman-Narici, Chapter 10], exercise 6, 8, 9

c. Prove [Bachman-Narici, Theorems 10.6 and 10.7].d. [Bachman-Narici, Chapter 9], exercises 14, 18.


a. [Bachman-Narici, Chapter 10], exercises 2, 3, 4, 10

25

Lecture 17: Riesz and Lax-Milgram representation theorems.



0:00 Riesz representation theorem. This is [Lax, Theorem 6.4].3:20 The null space of a non-trivial bounded linear functional (BLF) has co-

dimension 1. If two BLF have the same null space, then one is a constantmultiple of the other. This is [Lax, Lemma 6.5], (i) and (ii).

5:37 Proof of the first claim.10:00 Proof of the second claim.14:43 The null space of a BLF is a closed subspace.16:40 Proof of the Riesz representation theorem.27:44 Lax-Milgram representation theorem. Let B : X × X → K be a function

such that(a) For each fixed y, B(x, y) is a linear function of x and, for each fixed

x, B(x, y) is a a skewlinear function of y.(b) B is bounded in the sense that there is a constant C1 such that|B(x, y) | ≤ C1 ‖x‖ ‖y‖ for all x, y in X.

(c) There is a constant c0 > 0 such that |B(z, z) | ≥ c0 ‖z‖2 for all z inX.

Let ` be a BLF on X. Then, there exists a unique y ∈ X such that`(x) = B(x, y) for all x ∈ H. This is [Lax, Theorem 6.6]

31:50 Proof of Lax-Milgram theorem. Step 1. For all x ∈ X, there exists T (x) ∈X such that B(y, x) = 〈y , T (x)〉 for all y ∈ X. T is linear.

37:14 Step 2: The set T (x) : x ∈ X is a CLS of X.45:40 Step 3: We have that T (x) : x ∈ X = X.48:55 Step 4: Conclusion of the proof.

Further readings.

A. Sections 2.10–2.15 of [Zeidler-108] and [Lax, Chapter 7] provide examplesof applications of the Riesz and the Lax-Milgram representation theorems.Some will be examined in lectures 19.

26

Lecture 18: Orthonormal sets and closed linear spans.

Summary. This lecture is based on [Lax, Section 6.4]. The statements of thissection are sometimes imprecise. It’s never clear if the set of indices may be un-countable. The lecture clarifies this point.


0:00 Recollection of the linear span and the closed linear span of a set. Below,the set I may be uncountable.

4:20 Proposition: Let X be a HS and Y be the closed linear span (CLS) ofxθ : θ ∈ I. Then, z ∈ Y if and only if the following property is in force.If x ∈ X is such that 〈x, xθ〉 = 0 for all θ ∈ I, then 〈x, y〉 = 0. This is [Lax,Theorem 6.7].

13:50 Definition of an orthonormal family xθ : θ ∈ I and definiton of basis.16:15 Remark: Given an orthonormal set xj : j ≥ 1 and αj : j ≥ 1, αj ∈ K,

such that∑j |αj |2 <∞, then

∑j αj xj is well defined as an element of X .

21:06 Lemma (Bessel’s inequality) : Let xθ : θ ∈ I be an orthonormal set andx ∈ X. Then, (a) the set θ ∈ I : 〈x, xθ〉 6= 0 is at most countable. (b)∑θ∈I | 〈x, xθ〉 |2 ≤ ‖x‖2. This last sum is well defined because at most a

countable number of terms are different from 0.23:16 Proof of Bessel’s inequality. Step 1. Let J = θk : k ≥ 1 ⊂ I be countable.

Then,∑k≥1 | 〈x, xθk〉 |2 ≤ ‖x‖2.

29:16 Step 2. The set Jm = θ ∈ I : | 〈x, xθk〉 | > 1/m is finite. Assertion (a)follows

33:52 Step 3. Proof of assertion (b).35:47 Proposition: Let xθ : θ ∈ I be an orthonormal set. Then, the CLS of

xθ : θ ∈ I is the set ∑j≥1

αj xθj : θj ∈ I , αj ∈ K ,∑j≥1

|αj |2 <∞.

This is [Lax, Theorem 6.8]. Proof that this set is linear, closed and containsall xθ.

53:15 Proof of the converse.56:03 Remark: Let xθ : θ ∈ I be an orthonormal set, and x =

∑j≥1 αj xθj

such that∑j≥1 |αj |2 <∞. Then, ‖x‖2 =

∑j≥1 |αj |2, and αj = 〈x, xθj 〉.

Further readings.

A. Section 3.1 of [Zeidler-108] present some of these results from another per-spective.

*B. Section 3.2 of [Zeidler-108] apply these results to Fourier series. This is avery important application. I recommend reading this section.

*C. Section 9.4 of [Bachman-Narici] provides an example where the set I isuncountable.


*a. Chapter 3 of [Zeidler-108], exercise 1.*b. Chapter 9 of [Bachman-Narici], exercises 4, 5, 6, 9

c. Chapter 9 of [Bachman-Narici], exercise 3

27

Lecture 19: Orthonormal bases.



0:00 Theorem. All H. S. contain an orthonormal basis. This is [Lax, Theorem6.9].

8:25 Gram-Schmidt. Let X be a H.S., A = N or A = 1, . . . ,m, and xp :p ∈ A be a set of l. i. elements. Then, there exists a set yp : p ∈ A oforthonormal vectors such that for all k ≥ 1, LS of xp : 1 ≤ p ≤ k is equalto LS of yp : 1 ≤ p ≤ k.

11:30 Proof of Gram-Schmidt.22:50 Lemma: every basis has the same cardinality. This results permits to define

the dimension of a HS. A similar proof can found in [Bachman-Narici,Theorem 10.2].

29:27 Parseval identity. Let xθ : θ ∈ I be an orthonormal basis of a HS. Then,for all x ∈ X, x =

∑θ∈I〈x, xθ〉xθ and ‖x‖2 =

∑θ∈I | 〈x, xθ〉 |2. In these

sums, at most a countable number of terms are non-zero.

Further readings.

A. Chapter 10 of [Bachman-Narici] proves most of the results presented in thislecture.

B. Sections 3.1 and 3.3 of [Zeidler-108] state the Parseval identity and theGram-Schmidt orthogonalizaion method.

C. Sections 3.2 and 3.4 of [Zeidler-108] present applications to Fourier seriesand apply the Gram-Schmidt orthogonalizaion method to polynomials inL2.


*a. Show that a Hilbert space is separable if and only if it admits a countableorthonormal basis.

*b. Chapter 6 of [Lax], exercises 8, 9, 10.*c. Chapter 3 of [Zeidler-108], exercise 2.d. Prove [Bachman-Narici, Theorems 10.1, 10.3 and 10.4]e. Chapter 10 of [Bachman-Narici], exercise 5f. Chapter 9 of [Bachman-Narici], exercises 12, 13, 16.


a. Chapter 9 of [Bachman-Narici], exercises 7, 8, 21.b. Chapter 10 of [Bachman-Narici], exercises 1c. Chapter 3 of [Zeidler-108], exercises 3, 4, 5, 6, 7 (concern properties of

Fourier transforms and completeness in L2 of sequences of functions).

28

Lecture 19A: A quadratic variational problem

Summary. This lecture is based on [Zeidler-108, Section 2.4].


0:00 Formulation of the problem, Let G be a domain of Rd and let f , g : Rd → Rbe two continuous functions. We want to find u ∈ C2(G)∩C(G) such that−∆u = f G

u = g ∂G

6:33 Consider the quadratic variational problem on a Hilbert space X over thereal numbers:

inf

(1/2)a(x, x) − b(x) : x ∈ X,

where a is bilinear, bounded, symmetric and coercive, and b is bounded andlinear. This problem has a unique solution, This is [Zeidler-108, Theorem2.A]. Recall that coercive means that there exists c0 > 0 such that a(x, x) ≥c0‖x‖2 for all x ∈ X.

11:08 Proof of the quadratic variational problem.34:00 Corollary: A vector x0 is a solution of the quadratic variational problem if

and only if a(x0, y) = b(y) for all y ∈ X.

29

Lecture 19B: The Dirichlet principle.



0:00 Statement of three versions of the boundary-value problem for the Poissonequation. Let G ⊂ Rd be a domain.

Version 1. Consider the differential equation−∆u = f on G

u = g on ∂G

Version 2. For all test function ω ∈ C∞0 (G),

d∑j=1

∫G

(∂xju)(x) (∂xjω)(x) dx =

∫G

f(x)ω(x) dx

and u = g on ∂G.Version 3. Denote by H(u) the energy of a function u : G→ R, defined by

H(u) =1

2

d∑j=1

∫G

[(∂xj

u)(x)]2dx

Solve the variational problem

infH(u) −

∫G

f(x)u(x) dx : u ∈ C1(G) , u = g on ∂G.

This problem is called the Dirichlet problem.10:09 Lemma: Assume that u ∈ C2(G) is a solution of the Dirichlet problem.

Then. ∆u = f on G. This is Proposition 1(i) of [Zeidler-108, Chapter 2].12:55 Proof of the lemma.

30

Lecture 19C: Generalized derivatives and Sobolev spaces.



0:00 Definition of the generalized derivative ∂xju of a function u ∈ L2(G).

3:50 ∂xju is uniquely defined in L2(G). This is Proposition 4 of [Zeidler-108,Chapter 2].

6:48 Examples of generalized derivatives in one dimension.11:34 The space W 1

2 (G) and the scalar product 〈u , v〉1,2.17:34 Proposition: The space W 1

2 (G) is a Hilbert space. This is Proposition 7 of[Zeidler-108, Chapter 2].

23:04 The Hilbert subspaceW 1

2 (G). See Proposition 9 of [Zeidler-108, Chapter2].

25:13W 1

2 (G) is “the subspace of functions in W 12 (G) which vanish at the bound-

ary”. Lemma (d = 1): Let u ∈W 1

2 ((a, b)). Then, there exists v ∈ C([a, b])such that u = v a.s. and v(a) = v(b) = 0.

36:05 Theorem: Assume that G is bounded and ∂G is smooth. There exists aconstant C0 such that∫

∂G

u(x)2 σ(dx) ≤ C0 ‖u‖21,2

for all u ∈W 12 (G). Note that on the left-hand side, the integral is performed

with respect to the surface measure on ∂G. Proof in the case where G is arectangle for u in C1(G).

49:09 Extension to functions u in W 12 (G). If u ∈

W 1

2 (G), then∫∂G

u(x)2 σ(dx) =0.

31

Lecture 19D: The Poincare inequality.

Summary. .


0:00

32

Lecture 20: Uniform boundedness principle.

Summary. This lecture is based on [Royden, Section 7.7].


0:00 Definition of Baire spaces. Baire category theorem: A complete metricspace is a Baire space.

1:41 Proof of the theorem.18:00 Theorem: Let X be a complete metric space, and let fα : X → R, α ∈ I,

be a family (may be uncountable) of continuous functions, Assume that forall x ∈ X, there exists M(x) < ∞ such that supα∈I | fα(x) | ≤ M(x).Then, there exist an open set G ⊂ X and a finite constant C such thatsupα∈I | fα(x) | ≤ C0 for all x ∈ G.

20:50 Proof of this theorem.

Further readings.

A. The reader will find in [Royden, Section 7.7] the definition of sets of firstand second category and the statement of Baire’s category theorem.

B. [Zeidler-109, Section 3.1] presents examples of sets of first and second cate-gory. [Zeidler-109, Section 3.2] applies Baire category theorem to prove theexistence of a continuous function which is nowhere differentiable.

C. Section 0.2 of [Yosida] presents some further consequences of the Bairecategory theorem.


a. Prove Proposition 1 of [Zeidler-109, Section 3.2].b. [Royden, Chapter 7], problems 27, 29, 30, 31.c. [Zeidler-109, Chapter 3], problems 1, 2.


a. [Royden, Chapter 7], problems 24, 25, 26, 28.b. [Zeidler-109, Chapter 3], problem 3.

33

Lecture 21: Weak convergence.



0:00 Definition of weak convergence of a sequence (xn : n ≥ 1) in a NLS. Strongconvergence implies weak convergence,

3:00 The converse does not hold: an example in `2.8:20 A second example in the space C([0, 1]).

Further readings.

A. [Zeidler-109, Section 2.4] provides further examples. The reader will find inthis section the proof that in finite-dimensional spaces, weak convergenceimplies strong convergence.

B. [Bachman-Narici, Section 14.1].


*a. [Bachman-Narici, Chapter 14], exercises 2, 9, 10.*b. Prove [Lax, Chapter 10], exercise 2. This is a very nice result.*c. Fill the details of Standard example 3 of [Zeidler-109, Section 3.3].d. Prove Theorem 2.C and Corollary 4 in [Zeidler-109, Section 2.4].e. Prove Proposition 3 in [Zeidler-109, Section 2.4].f. [Zeidler-109, Chapter 3], problems 5, 7, 16, 17.


a. [Bachman-Narici, Chapter 14], exercise 1 and 3.

34

Lecture 22: Uniform boundedness of weak converging sequences.



0:00 Lemma: Let X be a NLS and (xn : n ≥ 1) a sequence such that (a) Thereexists C0 <∞ such that ‖xn‖ ≤ C0 for all n ≥ 1, and (b) `(xn)→ `(x) forall ` in a dense subset of X ′. Then, xn x (xn converges weakly to x).This is Theorem 1 of [Lax, Chapter 10]

5:43 Theorem: Let X be a BS, and fα : α ∈ I a set of continuous functionswhich are sub-additive, absolutely homogeneous and pointwisely bounded.Then, there exists C0 <∞ such that supα |fα(x) | ≤ C0 ‖x‖ for all x ∈ X.This is Theorem 2 of [Lax, Chapter 10]. (We need X to be a Banach spacebecause Baire’s category theorem requires the metric space to be complete.)

20:00 Corollary: Let X be a BS, and `α : α ∈ I a set of BLF. Suppose that forall x ∈ X, there exists C(x) < ∞ such that supα |`α(x)| ≤ C(x). Then,there exists C0 < ∞ such that supα ‖ `α ‖ ≤ C0. This is Theorem 3 of[Lax, Chapter 10].

27:04 Corollary: Let X be a NLS, and xα : α ∈ I a subset of X. Suppose thatfor all ` ∈ X ′, there exists C(`) <∞ such that supα |`(xα)| ≤ C(`). Then,there exists C0 < ∞ such that supα ‖xα ‖ ≤ C0. This is Theorem 4 of[Lax, Chapter 10].

35:13 Corollary: Let X be a NLS, and (xn : n ≥ 1) a weakly converging sequenceof elements of X (xn x for some x ∈ X). Then, there exists C0 < ∞such that supn ‖xn ‖ ≤ C0. This is Theorem 4’ of [Lax, Chapter 10].

39:07 Proposition: Let X be a NLS, and (xn : n ≥ 1) a sequence of elements ofX which converges weakly to x, xn x for some x ∈ X. Then, ‖x‖ ≤lim inf ‖xn‖. This is Theorem 5 of [Lax, Chapter 10]. The inequality canbe strict. For example, consider a sequence which converges weakly to 0but whose norm is constant equal to 1.

Further readings.

A. [Zeidler-109, Section 3.3] presents the uniform boundedness theorem inthe context of operators. [Zeidler-109, Section 3.4] applies the uniformboundedness principle to the cubature formula (an approximation of one-dimensional integrals of continuous functions by Riemann sums)


a. Prove Theorem 3.B and Corollary 1 of [Zeidler-109, Section 3.3].b. Prove Proposition 2 of [Zeidler-109, Section 3.3].c. Prove Proposition 6 and Corollary 7 of [Zeidler-109, Section 2.8].d. [Zeidler-109, Chapter 3], problem 4.

35

Lecture 23: Weak sequentially compactness.



0:00 Definition of weak sequentially compactness subsets. These sets are bounded.4:47 Theorem. Assume that the BS X is reflexive. Then, the ball of radius 1 is

weakly sequentially compact. This is Theorem 7 of [Lax, Chapter 10].6:40 Proof in the case where X is separable.

28:14 Proof in the general case.36:18 Remark: Recall that the ball of radius one is not compact for the strong

topology.


a. Prove Theorem 2.D and Corollary 2 of [Zeidler-109, Section 2.5].b. Prove Proposition 4 of [Zeidler-109, Section 2.5].

36

Lecture 24: Weak∗ topology.

Summary. This lecture is based on [Lax, Section 10.5]. Recall from Lecture 14Athat the dual space of M = C([−1, 1]) is the space of finite, Borel signed measuresendowed with the norm ‖ ν ‖ = ν+([−1, 1]) + ν−([−1, 1]), where ν+, ν− forms theHahn-Jordan decomposition of ν. See the references given in that lecture on thedual of X, that is, the bidual of M .


0:00 Let M be a BS and X = M ′. Definition of weak∗-convergence in X. Thisconvergence is weaker than the weak convergence.

5:33 If M is reflexive, then weak∗-convergence in X implies weak convergence.9:03 Example: X the BS of finite signed measures on [−1, 1]. Here and through-

out this lecture, by signed measures I mean finite signed measures.15:00 An example to illustrate that weak∗-convergence does not imply weak con-

vergence.16:08 Actually, what we need here is that the space of bounded, Borel mea-

surable functions defined on the interval [−1, 1], denoted by L∞([−1, 1]), iscontained in X ′ = M ′′. This is clear, since for f ∈ L∞([−1, 1]), Lf : X → Rdefined by

Lf (x) =

∫ 1

−1f dx .

is a bounded linear functional. For references on the description of X ′ =M ′′, see Lecture 14A.

23:28 Proposition: If xn converges to x in the weak∗ topology, then supn ‖xn‖ <∞. This is [Lax, Theorem 10.11].

26:36 Remark: If xn converges to x in the weak∗ topology, then ‖x‖ ≤ lim infn ‖xn‖.It is not correct that there exists m ∈M , ‖m‖ = 1, such that ‖x‖ = |x(m)|.Here is the argument. Fix ε > 0. By definition of ‖x‖, there existsm ∈ M , ‖m‖ = 1, such that ‖x‖ ≤ |x(m)| + ε. As xn weak∗ convergesto x, |x(m)| = limn |xn(m)| ≤ ‖m‖ lim infn ‖xn‖ = lim infn ‖xn‖ because‖m‖ = 1. This completes the proof.

29:23 Definition of w∗-sequentially compact subset.30:45 Theorem: Let M be a separable BS and X = M ′. Then, the closed ball of

radius 1 in X is w∗-sequentially compact. This is [Lax, Theorem 10.12].

Further readings.

A. [Yosida, Section V.1].


*a. Prove [Yosida, Chapter V], Theorems 1 – 5 and 7 – 10.*b. [Zeidler-109, Chapter 3], problem 6.

37

Lecture 25: Applications of weak convergence.

Summary. This lecture is based on Sections 11.1 and 11.2 of[Lax].


0:00 Example: Let (fn : n ≥ 1) be a sequence in C([−1, 1]). We present nec-essary and sufficient conditions for the weak∗ convergence in C([−1, 1])′ ofµn(dt) = fn(t)dt to the Dirac measure δ0. This is [Lax, Theorem 11.1].

46:34 Theorem: There exists a periodic, continuous function f : R → R whoseFourier series diverges at one point. This is [Lax, Theorem 11.2].

38

Lecture 26: Bounded linear operators.



0:00 Definition of bounded linear maps M : X → Y , where X, Y are BS.3:43 Lemma: A bounded linear map (BLM) M is bounded if and only if it is

continuous. This is [Lax, Theorem 15.1].9:36 Let X, Y be NLS, and M : X → Y a BLM. Denote by X, Y the completion

of X, Y , respectively. Then, there exists a unique BLM M : X → Y suchthat M(x) = M(x) for all x ∈ X. Prove that the extension is unique.

28:58 Definition of the norm of a bounded linear map M .35:10 Let X, Y be BS. Then, L(X,Y ) = M : X → Y |M is a BLM is a linear

space. ‖M‖ is a norm in L(X,Y ). This is [Lax, Theorem 15.2].43:45 Let X, Y be BS. Then, L(X,Y ) is a BS. This is [Lax, Theorem 15.3].59:03 Remark. Let X, Y be BS, and M ∈ L(X,Y ). Then, the null space N =

x ∈ X : Mx = 0 is a closed linear space (CLS) of X.1:02:54 Lemma: Let X, Y be BS, M ∈ L(X,Y ), and N = x ∈ X : Mx = 0.

Then, M0 : X∣∣N → Y defines a one-to-one linear map such that ‖M0‖ =

‖M‖. Moreover, the range of M and M0 coincide. This is [Lax, Theorem15.4].

Further readings.

A. Sections 5, 9, 20 and 25 of [Zeidler-108, Chapter 1].B. [Bachman-Narici, Section 14.2].


*a. Prove Propositions 3 and 6 of [Zeidler-108, Section 1.20].b. Prove Proposition 3 of [Zeidler-108, Section 1.9]c. Fill the details of Standard Example 4 [Zeidler-108, Section 1.20]d. [Zeidler-108, Chapter 1], problems 3 and 4. The definition of the spectrum

of a linear operator is given in Section 1.25.e. Prove Theorems 8 and 9 of [Bachman-Narici, Chapter 14].f. [Bachman-Narici, Chapter 14], exercises 4, 6, 15, 16.g. [Zeidler-109, Chapter 3], problem 8.


a. [Zeidler-108, Chapter 1], problem 1j.b. [Bachman-Narici, Chapter 14], exercises 5, 7, 14.

39

Lecture 27: Transpose of Bounded Linear Operators.



0:00 Let X, Y be normed linear spaces (NLS). The composition of a boundedlinear operator (BLO, a nomenclature equivalent to bounded linear map)M : X → Y with a bounded linear functional (BLF) ` : Y → K: m = ` Mbelongs to X ′.

2:28 The transpose operator M ′ : Y ′ → X ′ is defined by M ′` = `M is a BLO.Moreover, ‖M ′‖ ≤ ‖M‖. This operator is sometimes called the dual or theconjugate operator.

8:09 Let x ∈ X, m ∈ X ′. We denote m(x) as X(x,m)X′ or (x,m), for short.Hence, (Mx, `) = (x,M ′`) for all x ∈ X, ` ∈ Y ′ and BLO M : X → Y .

10:52 Lemma: ‖M ′‖ = ‖M‖. This result and the ones below are part of Theorem5 of [Lax, Chapter 15].

16:30 Remark: Let M1, M2 ∈ L(X,Y ), α ∈ K, then (αM1 +M2)′ = αM ′1 +M ′219:20 Lemma: Denote by RM the range of M and by NM ′ the null space of M ′.

Then, R⊥M = NM ′ .20:23 NM ′ = ` ∈ Y ′ : M ′(`) = 0. Replace M ′(`′) = 0 by M ′(`) = 0.28:06 Lemma: NM = R⊥M ′ .29:30 Note that there is a slight abuse of notation here. Indeed, since RM ′ ⊂ X ′,

by the original definition, R⊥M ′ should represent be a subset of X ′′. Instead,according to the definition given, it is a subset of X.

38:12 The argument could be smoother. Assume that X is a HS, and let M : X →X be a BLO. DefineM∗ : X → X as follows. Fix y ∈ X, and let Ly : X → Kbe given by Ly(x) :=< Mx, y >. Ly is a BLF. By Riesz’ representationtheorem, there exists zy ∈ X such that Ly(x) =< x, zy > for all x ∈ X.The map y 7→ zy is bounded and linear. Denote this map by M∗ so thatM∗y = zy ∈ X and

< Mx, y > = Ly(x) = < x, zy > = < x,M∗y >

for all x, y in X. This map is called the adjoint of M .On the other hand, denote by `y : X → K, y ∈ X, the BLF given by

`y(x) =< x, y >. Then, with the notation introduced above,

(x,M ′`y) = (M ′`y)(x) = `y(Mx) = < Mx, y > = < x,M∗y > .

In conclusion, in the case of a HS, we obtained a BLM M∗ : X → X suchthat for all x, y ∈ X,

(x,M ′`y) = < x,M∗y > .

On the left-hand side we have the product X(·, ·)X′ , while on the right-handside the usual scalar product in X.

Further readings.

A. [Zeidler-109, Section 3.10].B. [Zeidler-108, Section 5.2] for the definition of adjoint operators.C. See [Zeidler-109, Section 3.11] for exacteness of the duality functor.D. [Bachman-Narici, Sections 17.3 and 17.4].

40


*a. Prove [Zeidler-109, Section 3.10], proposition 2.*b. Compute the dual and the adjoint operators in the finite-dimensional case.

See [Zeidler-109, Section 3.10], example 3.*c. Prove [Zeidler-109, Section 3.10], Proposition 5, Corollary 6 and Proposi-

tion 7.*d. [Lax, Chapter 15], exercises 3 and 4.

e. [Zeidler-109, Chapter 3], problems 14, 15.f. [Bachman-Narici, Chapter 17], exercise 4.


a. [Bachman-Narici, Chapter 17], exercises 5 and 6.

41

Lecture 28: Strong and weak convergence of operators.

Summary. This lecture is based on [Lax, Section 15.2]. Throughout, X and Yare BS.


0:00 Definition of uniform, strong and weak topologies in L(X,Y ).4:30 Definition: A sequence (Mn : n ≥ 1) of BLO strongly converges if for all

x ∈ X, Mnx ∈ Y converges strongly in Y . Lemma: If (Mn : n ≥ 1) stronglyconverges, then there exists a BLO M ∈ L(X,Y ) such that Mn convergesstrongly to M [For all x ∈ X, Mnx converges strongly to Mx in Y ]. This

is represented by Mns→M .

7:17 Proof of the previous result.9:48 and 10:59 The correct terminology for the property fn(αx) = |α| fn(x)

is absolutely homogeneous and not positive homogeneous as I used in thelecture.

17:55 Definition: A sequence (Mn : n ≥ 1) of BLO weakly converges if for allx ∈ X, Mnx ∈ Y converges weakly in Y . Lemma: If (Mn : n ≥ 1) weaklyconverges, then there exists a BLO M ∈ L(X,Y ) such that Mn convergesweakly to M [For all x ∈ X, Mnx converges weakly to Mx in Y ]. This is

represented by Mn M or by Mnw→M .

24:25 Lemma: Assume that X is reflexive and that Mn M , then M ′n M ′.This is exercise 6 of [Lax, Chapter 15].

33:08 Claim: The previous statement is false if weak convergence is replaced bystrong convergence.

43:26 Theorem: Assume that (Mn : n ≥ 1) is bounded (‖Mn‖ ≤ C0 for somefinite constant C0) and that there exists a dense subset D of X such thatMnx converges strongly for all x ∈ D. Then, there exists M ∈ L(X,Y )

such that Mns→M . This is Theorem 6 of [Lax, Chapter 15].

46:16 Proof of the theorem.

Further readings.

A. [Bachman-Narici, Chapter 15].


*a. [Lax, Chapter 15], exercises 5 and 7(b).*b. Prove that uniform convergence implies strong convergence, which in turn

implies weak convergence.c. Prove [Bachman-Narici, Theorem 15.1].d. [Bachman-Narici, Chapter 15], exercises 2, 5.


a. [Bachman-Narici, Chapter 15], exercises 1, 3, 4, 6, 7.

42

Lecture 29: Principle of uniform boundedness for maps.

Summary. This lecture is based on [Lax, Section 15.3]. Throughout, X, Y and Zare BS.


0:00 Theorem: Let Mα : α ∈ I be a set of BLO in L(X,Y ). Assume thatfor all x ∈ X and ` ∈ Y ′, there exists a finite constant C(x, `) such that| (Mαx, `) | ≤ C(x, `) for all α ∈ I. Then, there exists a constant C0 suchthat ‖Mα‖ ≤ C0 for all α ∈ I. This is Theorem 7 in [Lax, Chapter 15].

Next result follows from the proof of this theorem. Let Mα : α ∈ Ibe a set of BLO in L(X,Y ). Assume that for all x ∈ X, there exists afinite constant C(x) such that ‖Mαx ‖ ≤ C(x) for all α ∈ I. Then, thereexists a constant C0 such that ‖Mα‖ ≤ C0 for all α ∈ I. This is theBanach-Steinhaus theorem. Here Y needs not to be complete.

3:49 This is the corollary at time [27:04] in Lecture 22 or Theorem 4 in [Lax,Chapter 10].

6:00 absolutely homogeneous and not positive homogeneous.6:26 This is the theorem at time [5:43] in Lecture 22 or Theorem 2 in [Lax,

Chapter 10].7:55 Remark: Let (Mn : n ≥ 1) be a sequence in L(X,Y ) such that Mn M

for some M ∈ L(X,Y ). Then, there exists C0 such that ‖Mn‖ ≤ C0 for alln ≥ 1. This is Corollary 7’ in [Lax, Chapter 15].

11:04 Let M ∈ L(X,Y ), N ∈ L(Y,Z). Then, NM belongs to L(X,Z) and‖NM‖ ≤ ‖N‖ ‖M‖. This is Theorem 8 in [Lax, Chapter 15].

13:48 (NM)′ = M ′N ′.

Further readings.

A. [Zeidler-109, Section 3.3] covers part of the content of this lecture.B. [Zeidler-109, Section 3.4] provides an application of the results presented in

this lecture.C. [Bachman-Narici, Section 15.2] also covers part of the content of this lecture.


*a. [Lax, Chapter 15], exercises 8, 9 and 10.*b. [Bachman-Narici, Chapter 15], exercise 8.

c. Prove Theorem 3.B of [Zeidler-109, Section 3.3].d. Prove Theorem 15.3 of [Bachman-Narici, Section 15.3] and its corollary.e. Prove Theorem 15.4 of [Bachman-Narici, Section 15.3] and its corollary.f. Prove Theorem 15.5 of [Bachman-Narici, Section 15.3].g. [Bachman-Narici, Chapter 15], exercise 9, 10 and 13.


a. [Bachman-Narici, Chapter 15], exercise 11, 12.

43

Lecture 30: The open mapping principle.

Summary. This lecture is based on [Lax, Section 15.5]. Throughout, X and Yare BS.


0:00 Theorem: Let M ∈ L(X,Y ) be surjective. Then, there exists r > 0 suchthat B(0, r) ⊂ M B(0, 1). Here, B(x, r) represents the open ball of radiusr centered at x. This is Theorem 9 in [Lax, Chapter 15].

1:57 Remark. Let S be a Baire space and (Fn : n ≥ 1) be a family of closed setssuch that ∪n≥1Fn = S. Then, there exists n ≥ 1 and an open set G suchthat G ⊂ Fn.

9:36 Proof of the theorem. Step 1: There exists n ≥ 1 and an open set G ⊂ Ysuch that G ⊂MB(0, n).

13:20 Step 2: There exist m ≥ 1 and r > 0 such that B(0, r) ⊂MB(0,m).

18:22 Step 3: There exist s > 0 such that B(0, s/2k) ⊂MB(0, 1/2k) for all k ≥ 1.21:41 Conclusion of the proof.31:42 The open mapping theorem. Let M ∈ L(X,Y ) be surjective. Then, M

maps open set to open sets. This is Theorem 10 in [Lax, Chapter 15].38:47 Let M ∈ L(X,Y ) be bijective. Then, M−1 ∈ L(Y,X). This is Theorem 11

in [Lax, Chapter 15].

Further readings.

A. [Zeidler-109, Section 3.5] covers part of the content of this lecture.B. [Bachman-Narici, Chapter 16] covers this lecture and the next one.


*a. Prove [Zeidler-109, Section 3.5], Corollary 2.

44

Lecture 31: The closed graph theorem.

Summary. This lecture is based on [Lax, Section 15.5]. Throughout this lecture,X and Y are BS.


0:00 Definition of the graph of a BLO.2:12 Definition of a closed operator.4:54 Remark: A bounded operator is closed.6:30 Remark: If M is a closed operator, then its graph G is complete, that is, G

is a BS.9:12 Theorem: A closed operator is bounded. This is Theorem 12 in [Lax,

Chapter 15].13:23 We apply here Theorem 11 in [Lax, Chapter 15], see Lecture 30 at time

[38:47].16:50 Definition of compatible norms. The norms ‖ · ‖1 and ‖ · ‖2 are compatible

if xn‖ · ‖1→ x, xn

‖ · ‖2→ x, then x = y.18:24 Application 1: Assume that (X, ‖ · ‖1) and (X, ‖ · ‖2) are BS and that the

norms are compatible. Then, the norms are equivalent. This is Theorem13 in [Lax, Chapter 15].

21:14 Proof of the previous result.26:33 Assume that X = A ⊕ B, where A, B are closed subsets of X. Let PA :

X → A be defined by PAx = a where a is the unique element in thedecomposition x = a+ b, a ∈ A, b ∈ B. Then, (a) PA is linear Show it; (b)PAw = w for w ∈ A, PAz = 0 for z ∈ B; (c) P 2

a = PA, PAPB = 0.33:05 Application 2: PA is continuous. This is Theorem 15 in [Lax, Chapter 15].

Further readings.

A. Read the proof of Theorem 3E in [Zeidler-109, Section 3.12].B. [Bachman-Narici, Chapter 16]. In this reference and in [Zeidler-109, Section

3.7], the operator M is assumed to be defined on a subspace D(M) of X.C. [Zeidler-109, Section 3.7] Read the Example 2.D. [Zeidler-109, Section 3.9] examines more closely decompositions of the form

X = A⊕B, projections and codimension.


*a. [Lax, Chapter 15], exercises 11, 12 and 13.*b. Prove Example 3 in [Zeidler-109, Section 3.7]. The definition of self-adjoint

oeprator is given in [Zeidler-108, Section 5.2].*c. Prove Theorem 1 in [Bachman-Narici, Chapter 16].*d. In [Bachman-Narici, Chapter 16], read the example of a linear transforma-

tion that is closed but not bounded.*e. Prove Theorems 3, 4 in [Bachman-Narici, Chapter 16].

f. Read the proof of Theorem 5 in [Bachman-Narici, Chapter 16].g. [Bachman-Narici, Chapter 16], exercises 4, 5, 6, 7.

45


a. [Bachman-Narici, Chapter 16], exercises 1, 2, 3, 8, 9, 10.

46

Lecture 32: Examples of bounded linear maps: Integral operators.



0:00 The set-up of measurable spaces (Sj ,Bj , µj).4:11 The integral operator (Af)(s) =

∫S1K(s, t) f(t)µ1(dt).

5:52 Case 1: A is a bounded linear operator from L1(S1) to L∞(S2) if

sups∈S2,t∈S1

|K(s, t) | < ∞ .

10:29 Case 2: A is a bounded linear operator from L∞(S1) to L1(S2) if∫s∈S2

µ2(ds)

∫S1

|K(s, t) |µ1(dt) < ∞ .

14:36 Case 3: A is a bounded linear operator from L2(S1) to L2(S2) if∫S2

µ2(ds)

∫S1

K(s, t)2 µ1(dt) < ∞ .

18:54 Case 3’: A is a bounded linear operator from L2(S1) to L2(S2) if

sups∈S2

∫S1

|K(s, t) |µ1(dt) < ∞

and

supt∈S1

∫S2

|K(s, t) |µ2(ds) < ∞ .

Further readings.

A. [Zeidler-108, Sections 1.5, 1.11] provide examples of integral operators inwhich the kernel may depend on the function.


*a. Find sufficient conditions for A to be a bounded operator from L1 to L1

and from L∞ to L∞.*b. Find sufficient conditions for A to be a bounded operator from Lp to Lq,

1 < p, q <∞.*c. Compute the transpose of A and the adjoint of A in the case where S1 = S2,

µ1 = µ2 and A : L2(S1)→ L2(S1).*d. Prove Standard Example 12 in [Zeidler-108, Section 1.11]

47

Lecture 33: Symmetric operators.

Summary. This lecture is based on [Zeidler-108, Section 4.1]. Throughout thislecture, X is a HS over the complex field.


0:00 Definition of symmetric operators A : D(A) ⊂ X → X.3:13 Definition of eigenvalues and eigenfunctions of symmetric operators. Main

properties: (a) 〈Ax, x〉 ∈ R; (b) Let λ ∈ C be an eigenvalue, then λ ∈R. These assertions and the next two are the content of Proposition 2 in[Zeidler-108, Section 4.1].

8:46 If λ1 6= λ2 are eigenvalues and x1, x2 their associated eigenfunctions, then〈u1, u2〉 = 0.

11:40 Assume that xj : j ≥ 1 forms an orthonormal basis of X and that all xjare eigenfunctions. Let λj : j ≥ 1 be the associated eigenvalues, and µbe an eigenvalue. Then, µ ∈ λj : j ≥ 1.

17:52 Proposition: Assume that A is symmetric and bounded. Then,

‖A‖ = sup‖x‖=1

| 〈Ax, x〉 | .

This is Proposition 3 in [Zeidler-108, Section 4.1].

Further readings.

A. [Bachman-Narici, Chapter 18] introduces the adjoint operator and presentssymmetric operators in a more general context where D(A) needs not tobe dense. It is worth reading.


a. [Zeidler-108, Chapter 4], exercise 1.b. [Bachman-Narici, Chapter 18], exercises 1, 2, 6, 8.


a. [Bachman-Narici, Chapter 18], exercises 3, 4, 5, 7, 9, 10.

48

Lecture 34: Eigenvalues of compact symmetric operators.



0:00 Definition of compact operators A : M ⊂ X → Y . Note that A is notassumed to be linear or bounded.

3:59 Example of a compact operator: integral operators on C([a, b]).11:50 Compact, symmetric operators. A : X → X in a complex Hilbert space.16:47 Theorem 4A of [Zeidler-108, Section 4.2]: Let X be a complex Hilbert

space. Assume that X 6= 0 is separable. Let A : X → X be a compact,symmetric operator. Then,(a) There exists an orthonormal basis xj : j ≥ 1 of eigenfunctions;(b) Let λj be the eigenvalue associated to xj . Then λj ∈ R. Moreover, If

λj 6= λk, then 〈xj , xk〉 = 0;(c) The eigenvalues have finite multiplicity: the dimension of the eigenspace

associated to an eigenvalue λ is finite.(d) Assume that the dimension X is infinite. Then, either there is a finite

number of eigenvalues different from 0, or limn λn = 0.24:07 Assume that Ax = 0 ⇒ x = 0. Step 1: ∃x1 ∈ X such that ‖x1‖ = 1,

Ax1 = λx1 for some λ ∈ R and |λ| = ‖A‖.35:26 Assume that the dimension of X is infinite. Step 2: There exists an or-

thonormal set xj : j ≥ 1 of eigenfunctions such that 0 < · · · ≤ |λj | ≤|λj−1| ≤ · · · ≤ |λ1| = ‖A‖. Here, λj is the eigenvalue associated to theeigenfunction xj .

49:17 Step 3: λn → 0.53:13 Step 4: Ax =

∑j≥1 λj 〈x, xj〉xj for all x ∈ X.

1:02:52 Step 5: x =∑j≥1〈x, xj〉xj for all x ∈ X.

1:05:55 Step 6: The eigenvalues have finite multiplicity.1:12:13 Proof in the case where the dimension of X is finite.1:15:54 Proof in the general case (without the hypothesis that Ax = 0⇒ x = 0).

Further readings.

A. [Reed-Simon, Chapter VI.5].


*a. Fill the gaps left in the lecture.

49

Lecture 35: The Fredholm alternative.



0:00 Let X be a separable HS and A be a compact symmetric operator.4:23 Theorem: Fix λ 6= 0 and z ∈ X. Then, the equation (λ − A)x = z has a

solution if and only if z ∈ N⊥λ , where Nλ is the null space of the operatorλ − A: Nλ = y ∈ X : Ay = λy. This is Theorem 4B in [Zeidler-108,Section 4.3].

10:21 Proof of the theorem. Case 1: λ 6= λj for all j. Mind that λ ∈ C. Then,(λ−A)x = z has a unique a solution.

22:07 Case 1: λ 6= λj for all j. Then, (λ−A)x = 0 has a unique a solution x = 0.25:07 Conclusion: the theorem holds if λ 6= λj for all j.27:11 Proof of the theorem in the Case 2, where λ = λj for some j. If z ∈ N⊥λ ,

then the equation (λ−A)x = z has a solution.38:48 Proof of the theorem in the Case 2, where λ = λj for some j. If the equation

(λ−A)x = z has a solution, then z ∈ N⊥λ .42:30 Remark: Assume that λ 6= 0, λ 6= λj for all j ≥ 1. Then, the operator

(λ−A)−1 exists and is a bounded linear operator.51:09 Corollary (Uniqueness implies existence): Fix λ 6= 0, z ∈ X. Assume that

(λ−A)x = z has at most one solution. Then, the operator (λ−A)−1 existsand is a bounded linear operator. Moreover, x = (λ−A)−1z is a solution.This is Corollary 1 in [Zeidler-108, Section 4.3].

57:00 Let X be a BS over C and A ∈ L(X,X). Definition of the resolvent set ofA, represented by ρ(A), and of the spectrum of A, denoted by σ(A).

1:00:12 In the context of this lecture, let Ω = λ ∈ C : λ is an eigenvalue . Then,Ω ⊂ σ(A) and C \ [Ω ∪ 0] ⊂ ρ(A).

1:04:30 Corollary: Assume that 0 is not an eigenvalue. If dim X < ∞, then0 ∈ ρ(A), and if dim X = ∞, then 0 ∈ σ(A). This is Corollary 2 in[Zeidler-108, Section 4.3].

1:12:29 Proof of the corollary in the case dim X =∞.

Further readings.

A. [Yosida, Chapter VII].

50

Lecture 36: An application to integral operators.

Summary. This lecture is based on This lecture is based on [Zeidler-108, Section4.4].


0:00 Consider the real HS X = L2([a, b]) where −∞ < a < b <∞.3:33 Lemma: Assume that K : [a, b]2 → R is continuous. Then, (Ax)(t) =∫ b

aK(t, s)x(s) ds is a compact BLO and Ax ∈ C([a, b]) for all x ∈ X. This

is Lemma 3 in [Zeidler-108, Section 4.4].10:40 Proof that Ax ∈ C([a, b]) for all x ∈ X.14:01 Proof that A is compact.23:52 Lemma: Assume that K(s, t) = K(t, s), Then, A is symmetric. This is

Lemma 3(c) in [Zeidler-108, Section 4.4].27:14 Lemma: Let yj : j ≥ 1 be an orthonormal basis of eigenfunctions of A.

Fix x ∈ X such that that x = Az for some z ∈ X. Then, ∀ ε > 0 ∃n0 suchthat for all m, n ≥ n0,

supt∈[a,b]

m∑j=n+1

| 〈x, yj〉 yj(t) | ≤ ε .

This is Corollary 2 in [Zeidler-108, Section 4.4]. This assertion is to becompared with the convergence in X of the sequence

∑j≤n〈x, yj〉 yj(t) to

x. The hypothesis that x = Az yields the convergence of the sums of theabsolute values, uniformly in t.

36:33 Proof of the Lemma.46:50 Proposition: Fix λ 6= 0, z ∈ X, and let Ω = λ ∈ R : λ is an eigenvalue .

(a) If λ 6∈ Ω, then the equation (λ− A)x = z has a unique solution givenby x = (λ−A)−1z.

(b) If λ ∈ Ω, then the equation (λ−A)x = z has a solution if and only ifz ∈ N⊥λ .

(c) Assume that z ∈ C([a, b]) and that x is a solution of (λ − A)x = z,then x ∈ C([a, b]).

This is Proposition 4 in [Zeidler-108, Section 4.4].

51

References

[Bachman-Narici] G. Bachman, L. Narici, Functional Analysis, Academic Press, New York, Lon-don 1966.

[Davis] M. Davis, A First Course in Functional Analysis. Notes on mathematics and its applica-

tions, Gordon and Breach, 1967. ISSN 0888-6113[Dudley] R. M. Dudley, Real Analysis and Probability, Cambridge studies in advanced mathemat-

ics, volume 74, Cambridge University Press, ISBN 0-521-80972-X, 2004.

[Dunford-Schwartz] N. Dunford, J. Schwartz, Linear Operators. Part 1: General Theory. Inter-science Publishers, Inc., New York. ISBN 0-470-22605-6

[Gilbarg-Trudinger] D. Gilbarg,N. S. Trudinger, Elliptic partial differential equations of secondorder. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

xiv+517 pp. ISBN: 3-540-41160-7 35-02

[Lax] P. D. Lax, Functional Analysis, Wiley-Interscience, ISBN 0-471-55604-1, 2002.[Oliveira] C. R. de Oliveira, Introducao a analise funcional, Projeto Euclides, IMPA, 2012 ISBN:

852440311X, 9788524403118.

[Reed-Simon] M. Reed, B. Simon, Methods of modern mathematical physics. I. Functional anal-ysis. Academic Press, New York-London, 1972. xvii+325 pp.

[Royden] H. L. Royden Real Analysis. Third edition. Macmillan Publishing Company, New York,

1988. xx+444 pp. ISBN: 0-02-404151-3[Taylor] S. J. Taylor, Introduction to Measure and Integration, Cambridge University Press, 1973.

ISBN 978-0-521-09804-5

[Yosida] K. Yosida Functional Analysis. Reprint of the sixth (1980) edition. Classics in Mathe-matics. Springer-Verlag, Berlin, 1995. xii+501 pp. ISBN: 3-540-58654-7

[Zeidler-108] E. Zeidler, Applied Functional Analysis: applications to mathematical physics, SeriesApplied mathematical sciences, volume 108, Springer-Verlag, ISBN 0-387-94442-7, 1995.

[Zeidler-109] E. Zeidler, Applied Functional Analysis, main principles and their applications, Se-

ries Applied mathematical sciences, volume 109, Springer-Verlag, ISBN 0-387-94422-2, 1995.

Documents

February 24, 2021w3.impa.br/~landim/Cursos/AF.pdfin any standard textbook on metric spaces. In portuguese, I recommend the proof of [Oliveira, Teorema 2.5]. Further readings. A.[Zeidler-108,