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WEAKER FORMS OF CONTINUITY AND VECTOR VALUED RIEMANN INTEGRATION. M.A.SOFI DEPARTMENT OF MATHEMATICS KASHMIR UNIVERSITY, SRINAGAR-190006 INDIA. - PowerPoint PPT Presentation
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M.A.SOFIDEPARTMENT OF MATHEMATICS
KASHMIR UNIVERSITY, SRINAGAR-190006INDIA
WEAKER FORMS OF CONTINUITY AND VECTOR VALUED RIEMANN
INTEGRATION
1.Classical Situation Given a continuous function ๐:แพ๐,๐แฟโโ, then a. f is Riemann integrable. b. f has a primitive: โ ๐น:แพ๐,๐แฟโโ such that F is differentiable on แพ๐,๐แฟ and ๐นโฒแบ๐กแป= ๐แบ๐กแป ๐๐ แพ๐,๐แฟ.
2.Banach spaces Let X be a Banach space and ๐:แพ๐,๐แฟโ๐ a continuous function. Then a. f is Riemann integrable. b. f has a primitive: โ ๐น:แพ๐,๐แฟโ๐ such that F is differentiable on แพ๐,๐แฟ and ๐นโฒแบ๐กแป= ๐แบ๐กแป ๐๐ แพ๐,๐แฟ. c. If ๐ is differentiable on [a, b], then ๐โฒ is Henstock integrable and ๐แบ๐ฅแป= ๐โฒแบ๐กแป๐๐ก,๐ฅ๐ โ๐ฅโแพ๐,๐แฟ.
3.Quasi Banach spaces Let X be a quasi Banach space. Then a. Continuity of ๐:แพ๐,๐แฟโ๐ does not imply Riemann integrability of f. b. Continuity โ Riemann integrability if and only if X is Banach. c. (Kalton) For X such that ๐โ= แบ0แป, continuity of f implies f has a primitive. (In particular, this holds for X= ๐ฟ๐แพ๐,๐แฟ,0 < ๐< 1). d. (Fernando Albiac) For X such that ๐โ is separating, there exists a continuous function ๐:แพ๐,๐แฟโ๐ failing to have a primitive.(In particular, for X =โ๐ , 0 < ๐< 1).
4. Riemann-Lebesgue Property (i) Definition: A Banach space X is said to have Riemann-Lebesgue(RL)- property if ๐:แพ๐,๐แฟโ๐ is continuous a.e on [a, b] if (and only if) it is Riemann integrable. (ii) Examples: a. (Lebesgue): โ has (RL)-property. Consequence: Finite dimensional Banach spaces have the (RL)-property. b. (G.C.da Rocha): โ1 has (RL)-property. c. (G.C.da Rocha):Tsirelson space. d. (G.C.da Rocha): Infinite dimensional Hilbert spaces do not have the (RL)-property. More generally, an infinite dimensional uniformly convex Banach does not possess (RL)-property.
(i) Definition: A Banach space X is said to have Weak Riemann-Lebesgue (WRL) - property if ๐:แพ๐,๐แฟโ๐ is weakly continuous a.e on [a, b] if (and only if) it is Riemann integrable. (ii) Theorem (Russel Gordon): C[0, 1] does not have (WRL)-property. (iii) Theorem (Wang and Yang): For a given measurable space แบฮฉ,ฦฉแป, the space ๐ฟ1(ฮฉ,ฦฉ)has (WLP).
(i) Theorem (Wang and Yang): For a given measurable space แบฮฉ,ฦฉแป, the space ๐ฟ1(ฮฉ,ฦฉ)has (WLP). As a generalisation of this result, we have: (ii) Theorem( E.A.Sanchez Perez et al): Let X be a Banach space having Radon-Nikodym property. Then the space ๐ฟ1(ฮฉ,ฦฉ,๐) of Bochner integrable functions has (WLP).
As a generalisation of this result, we have: (iv) Theorem( E.A.Sanchez Perez et al): Let X be a Banach space having Radon-Nikodym property. Then the space ๐ฟ1(ฮฉ,ฦฉ,๐) of Bochner integrable functions has (WLP).
5.Weaker forms of continuity: (i) Theorem (Wang and Wang): For a Banach space X, ๐:แพ๐,๐แฟโ๐ weakly continuous implies f is Riemann-integrable if and only if X is a Schur space(i.e., weakly convergent sequences in X are norm convergent). (ii) Theorem (V M Kadets): For a Banach space X, each weak*-continuous function ๐:แพ๐,๐แฟโ๐โ is Riemann-integrable if and only if X is finite dimensional.
6.Frechet space setting: (i) Definition: Given a Frechet space X, we say that a function ๐:แพ๐,๐แฟโ๐ is Riemann-integrable if the following holds: (*) โ ๐ฅโ๐ such that โ ๐> 0 and nโฅ 1, โ๐ฟ = ๐ฟ(๐,๐) > 0 such that for each tagged partition P= ๐ ๐,แพ๐ก๐โ1,๐ก๐แฟ,1 โค ๐ โค ๐ of [a, b] with ิก๐ิก= แบ๐ก๐ โ ๐ก๐โ1แป< ๐ฟ,1โค๐โค๐๐๐๐ฅ we have ๐๐แบ๐แบ๐,๐แปโ ๐ฅแป< ๐,
where, ๐แบ๐,๐แป is the Riemann sum of f corresponding to the tagged partition P= ๐ ๐,แพ๐ก๐โ1,๐ก๐แฟ,1 โค ๐ โค ๐ of [a, b] where ๐ = ๐ก0 < ๐ก1 < โฏ๐ก๐ = ๐ and ๐ ๐ โแพ๐ก๐โ1,๐ก๐แฟ,1 โค ๐ โค ๐. Here, ๐๐๐=1โ denotes a sequence of seminiorms generating the (Frechet)-topology of X. The (unique) vector x, to be denoted by เถฑ ๐แบ๐กแป๐๐ก๐๐ , shall be called the Riemann-integral of f over [a, b].
As a far reaching generalisation of Kadetโs theorem stated above, we have (i) Theorem (MAS, 2012): For a Frechet space X, each ๐โโvalued weakly*-continuous function is Riemann integrable if and only if X is a Montel space. (A metrisable locally convex space is said to be a Montel space if closed and bounded subsets in X are compact). Since Banach spaces which are Montel are precisely those which are finite dimensional, Theorem (ii) yields Kadetโs theorem as a very special case.
Ingredients of the proof : a. Construction of a โfatโ Cantor set. A โfatโ Cantor set is constructed in a manner analogous to the construction of the conventional Cantor set, except that the middle subinterval to be knocked out at each stage of the construction shall be chosen to be of a suitable length ๐ผ so that the resulting Cantor set shall have nonzero measure.
In the instant case, each of the 2๐โ1 subintervals ๐ด๐(๐) ( ๐ =1,2,โฆ,2๐โ1) to be knocked out at the kth stage of the construction from each of the remaining subintervals ๐ต๐(๐)( ๐ = 1,2,โฆ,2๐โ1) at the (k-1)th stage shall be of length ๐ผ= ๐(๐ด๐(๐)) = 12๐โ1 13๐, in which case ๐แ๐ต๐แบ๐แปแ= 12๐ (1โ ฯ 13๐๐๐=1 ) and, therefore, ๐แบ๐ถแป= 12.
b.Frechet analogue of Josefson-Nessenzwieg theorem: Theorem(Bonet, Lindsrtom and Valdivia, 1993): A Frechet space X is Montel if and only if weak*-null sequences in X* is strong*-null.
Sketch of proof: Necessity: This is a straightforward consequence of (b) above. Sufficiency: Assume that X is not Frechet Montel. By (b), there exists a sequence in X* which is weak*-null but not strong*-null. Denote this sequence by แผ๐ฅ๐โแฝ๐=1โ . Write ๐ด๐(๐) = [๐๐แบ๐แป,๐๐(๐)] and define a function ๐๐(๐):[0,1] โโ which is piecewise linear on ๐ด๐(๐) and vanishes off ๐ด๐(๐).
Put โ๐แบ๐กแป= ๐๐แบ๐แปแบ๐กแป,๐ก โแพ0,1แฟ,2๐โ1
๐=1 and define
๐แบ๐กแป= โ๐แบ๐กแป๐ฅ๐โ ,๐ก โแพ0,1แฟ.โ๐=1
Claim 1: f is weak*-continuous. This is achieved by showing that the series defining f is uniformly convergent in ๐๐โ.
Claim 2: f is not Riemann integrable. Here we use the fact that the Cantor set C constructed above has measure equal to 1 2เต and then produce a bounded subset B of X and tagged partitions ๐1 and ๐2of [0, 1] such that ๐๐ตเตซ๐แบ๐,๐1แปโ ๐แบ๐,๐2แปเตฏ> 1 2เต, where ๐๐ต is the strong*-seminorm on Xโ corresponding to B defined by ๐๐ตแบ๐แป= ax๐(๐ฅ)ax๐ฅโ๐ต๐ ๐ข๐ ,๐โXโ. This contradicts the Cauchy criterion for Riemann integrability of f.
We conclude with the following problem which appears to be open. PROBLEM 1: Characterise the class of Banach spaces X such that weakly*-continuous functions ๐:[๐,๐] โ๐ have a primitive F: ๐นโฒแบ๐กแป= ๐แบ๐กแป,โ๐ก โแพ๐,๐แฟ, i.e., ๐๐๐โ โ0แฅ๐นแบ๐ก+ โแปโ ๐น(๐ก)โ โ ๐(๐ก)แฅ= 0,โ๐ก โแพ๐,๐แฟ.
The following problem has a slightly different flavour and is motivated by the idea of โdecompositionโ of a โfinite dimensionalโ property, a phenomenon which has been treated in a recent work of the author โAround finite dimensionality in functional analysisโ (RACSAM, 2013). PROBLEM 2: Describe the existence of a locally convex topology ๐ on the dual of a Banach space X such that each ๐:แพ๐,๐แฟโ๐โ continuous w r t ๐ is Riemann integrable if and only if X is a Hilbert space.