ECE 602 Lumped Systems Theory October 21, 2007 1 ECE 602 Lecture Notes: Linear Spaces, Norms, and Inner Products 1. A linear space is defined as follows. Let F be a scalar field, e.g. the real numbers. Let X be a nonempty set. Given two mappings: addition mapping X × X into itself, and scalar multiplication mapping F × X into X such that the conditions below hold, X is called a linear space over F . (a) x 1 + x 2 = x 2 + x 1 ∀x 1 ,x 2 ∈ X (commutativity of addition) (b) x 1 +(x 2 + x 3 )=(x 1 + x 2 )+ x 3 ∀x 1 ,x 2 ,x 3 ∈ X (associativity of addition) (c) There is a unique element 0 of X such that 0 + x = x ∀x ∈ X . (existence of additive identity) (d) For every x ∈ X , there is a unique element -x of X such that -x + x = 0. (existence of additive inverse) (e) α(βx)=(αβ )x ∀α, β ∈ F, x ∈ X (associativity of scalar multiplication) (f) There is a unique element 1 ∈ F such that 1x = x ∀x ∈ X . (existence of multiplicative identity) (g) There is a unique element 0 ∈ F such that 0x =0 ∀x ∈ X . (existence of zero element) (h) α(x 1 + x 2 )= αx 1 + αx 2 ∀α ∈ F, x 1 ,x 2 ∈ X (distributivity of multiplication over addition) (i) (α + β )x = αx + βx ∀α, β ∈ F, x ∈ X (distributivity of addition over multiplication) 2. A norm is a real-valued function defined on a linear space X over a field F , satisfying the conditions below. For x ∈ X , x is called the norm of x. Nonnegativity x≥ 0 ∀x ∈ X . Definiteness x = 0 if and only if x = 0. Homogeneity αx = |α|x∀α ∈ F, x ∈ X . Triangle Inequality x + y≤x + y∀x, y ∈ X . 3. An inner product on a complex linear space X is a mapping from X × X to the field F of the complex numbers, satisfying the following. Positive Definiteness (x, x) > 0 for all nonzero x ∈ X . Additivity (x + y,z )=(x, z )+(y,z ) ∀x, y, z ∈ X . Symmetry (x, y)= (y,x) ∀x, y ∈ X . Homogeneity (αx, y)= α(x, y) ∀α ∈ F, x, y ∈ X .

ECE 602 Lecture Notes Linear Spaces, Norms, And Inner Products

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A linear space is defined as follows. Let F be a scalar field, e.g. the real numbers.Let X be a nonempty set. Given two mappings: addition mapping X × X intoitself, and scalar multiplication mapping F × X into X such that the conditionsbelow hold, X is called a linear space over F

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ECE 602 Lumped Systems Theory October 21, 2007 1

ECE 602 Lecture Notes:

Linear Spaces, Norms, and Inner Products

1. A linear space is defined as follows. Let F be a scalar field, e.g. the real numbers.Let X be a nonempty set. Given two mappings: addition mapping X X intoitself, and scalar multiplication mapping F X into X such that the conditionsbelow hold, X is called a linear space over F .

(a) x1 + x2 = x2 + x1 x1, x2 X (commutativity of addition)(b) x1 + (x2 + x3) = (x1 + x2) + x3 x1, x2, x3 X (associativity of addition)(c) There is a unique element 0 of X such that 0 + x = x x X. (existence of

additive identity)

(d) For every x X, there is a unique element x of X such that x + x = 0.(existence of additive inverse)

(e) (x) = ()x , F, x X (associativity of scalar multiplication)(f) There is a unique element 1 F such that 1x = x x X. (existence of

multiplicative identity)

(g) There is a unique element 0 F such that 0x = 0 x X. (existence of zeroelement)

(h) (x1 + x2) = x1 + x2 F, x1, x2 X (distributivity of multiplicationover addition)

(i) ( + )x = x + x , F, x X (distributivity of addition overmultiplication)

2. A norm is a real-valued function defined on a linear spaceX over a field F , satisfyingthe conditions below. For x X, x is called the norm of x.Nonnegativity x 0 x X.Definiteness x = 0 if and only if x = 0.Homogeneity x = ||x F, x X.Triangle Inequality x+ y x+ y x, y X.

3. An inner product on a complex linear space X is a mapping from X X to thefield F of the complex numbers, satisfying the following.

Positive Definiteness (x, x) > 0 for all nonzero x X.Additivity (x+ y, z) = (x, z) + (y, z) x, y, z X.Symmetry (x, y) = (y, x) x, y X.Homogeneity (x, y) = (x, y) F, x, y X.
ECE 602 Lumped Systems Theory October 21, 2007 2

Note that, if F is instead the reals, then the symmetry condition becomes (x, y) =(y, x) x, y X.

4. Comments

An inner product generates a norm x = (x, x)1/2.The Schwarz Inequality states that |(x, y)| xy x, y X.

5. Examples of Inner Products

(a) The usual inner product on Rn is (x, y) =n

i=1 xiyi.

(b) The usual inner product on Cn is (x, y) =n

i=1 xiyi.

(c) An inner product on the space of continuous complex-valued functions on theinterval [0, 1] can be defined by (x, y) =

10 x(t)y(t)dt.

(d) An inner product on the space of sequences of complex numbers satisfyingi=1 |xi|2

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