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 𝑓:ሾ𝑎,𝑏ሿ→ℝ, then a. f is Riemann integrable. b. f has a primitive: ∃ 𝐹:ሾ𝑎,𝑏ሿ→ℝ such that F is differentiable on ሾ𝑎,𝑏ሿ and 𝐹′ሺ𝑡ሻ= 𝑓ሺ𝑡ሻ 𝑜𝑛 ሾ𝑎,𝑏ሿ.

2.Banach spaces Let X be a Banach space and 𝑓:ሾ𝑎,𝑏ሿ→𝑋 a continuous function. Then a. f is Riemann integrable. b. f has a primitive: ∃ 𝐹:ሾ𝑎,𝑏ሿ→𝑋 such that F is differentiable on ሾ𝑎,𝑏ሿ and 𝐹′ሺ𝑡ሻ= 𝑓ሺ𝑡ሻ 𝑜𝑛 ሾ𝑎,𝑏ሿ. c. If 𝑓 is differentiable on [a, b], then 𝑓′ is Henstock integrable and 𝑓ሺ𝑥ሻ= 𝑓′ሺ𝑡ሻ𝑑𝑡,𝑥𝑎 ∀𝑥∈ሾ𝑎,𝑏ሿ.

3.Quasi Banach spaces Let X be a quasi Banach space. Then a. Continuity of 𝑓:ሾ𝑎,𝑏ሿ→𝑋 does not imply Riemann integrability of f. b. Continuity ⇒ Riemann integrability if and only if X is Banach. c. (Kalton) For X such that 𝑋∗= ሺ0ሻ, continuity of f implies f has a primitive. (In particular, this holds for X= 𝐿𝑝ሾ𝑎,𝑏ሿ,0 < 𝑝< 1). d. (Fernando Albiac) For X such that 𝑋∗ is separating, there exists a continuous function 𝑓:ሾ𝑎,𝑏ሿ→𝑋 failing to have a primitive.(In particular, for X =ℓ𝑝 , 0 < 𝑝< 1).

4. Riemann-Lebesgue Property (i) Definition: A Banach space X is said to have Riemann-Lebesgue(RL)- property if 𝑓:ሾ𝑎,𝑏ሿ→𝑋 is continuous a.e on [a, b] if (and only if) it is Riemann integrable. (ii) Examples: a. (Lebesgue): ℝ has (RL)-property. Consequence: Finite dimensional Banach spaces have the (RL)-property. b. (G.C.da Rocha): ℓ1 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 𝑓:ሾ𝑎,𝑏ሿ→𝑋 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 ሺΩ,Ʃሻ, the space 𝐿1(Ω,Ʃ)has (WLP).

(i) Theorem (Wang and Yang): For a given measurable space ሺΩ,Ʃሻ, the space 𝐿1(Ω,Ʃ)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 𝐿1(Ω,Ʃ,𝑋) 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 𝐿1(Ω,Ʃ,𝑋) of Bochner integrable functions has (WLP).

5.Weaker forms of continuity: (i) Theorem (Wang and Wang): For a Banach space X, 𝑓:ሾ𝑎,𝑏ሿ→𝑋 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 𝑓:ሾ𝑎,𝑏ሿ→𝑋∗ 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 𝑓:ሾ𝑎,𝑏ሿ→𝑋 is Riemann-integrable if the following holds: (*) ∃ 𝑥∈𝑋 such that ∀ 𝜀> 0 and n≥ 1, ∃𝛿 = 𝛿(𝜀,𝑛) > 0 such that for each tagged partition P= 𝑠𝑖,ሾ𝑡𝑖−1,𝑡𝑖ሿ,1 ≤ 𝑖 ≤ 𝑗 of [a, b] with ԡ𝑃ԡ= ሺ𝑡𝑖 − 𝑡𝑖−1ሻ< 𝛿,1≤𝑖≤𝑗𝑚𝑎𝑥 we have 𝑝𝑛ሺ𝑆ሺ𝑓,𝑃ሻ− 𝑥ሻ< 𝜀,

where, 𝑆ሺ𝑓,𝑃ሻ is the Riemann sum of f corresponding to the tagged partition P= 𝑠𝑖,ሾ𝑡𝑖−1,𝑡𝑖ሿ,1 ≤ 𝑖 ≤ 𝑗 of [a, b] where 𝑎 = 𝑡0 < 𝑡1 < ⋯𝑡𝑗 = 𝑏 and 𝑠𝑖 ∈ሾ𝑡𝑖−1,𝑡𝑖ሿ,1 ≤ 𝑖 ≤ 𝑗. Here, 𝑝𝑚𝑚=1∞ denotes a sequence of seminiorms generating the (Frechet)-topology of X. The (unique) vector x, to be denoted by න 𝑓ሺ𝑡ሻ𝑑𝑡𝑏𝑎 , shall be called the Riemann-integral of f over [a, b].

As a far reaching generalisation of Kadet’s theorem stated above, we have (i) Theorem (MAS, 2012): For a Frechet space X, each 𝑋∗−valued 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 Kadet’s theorem as a very special case.

Ingredients of the proof : a. Construction of a ‘fat’ Cantor set. A ‘fat’ 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 𝛼 so that the resulting Cantor set shall have nonzero measure.

In the instant case, each of the 2𝑘−1 subintervals 𝐴𝑘(𝑖) ( 𝑖 =1,2,…,2𝑘−1) to be knocked out at the kth stage of the construction from each of the remaining subintervals 𝐵𝑘(𝑖)( 𝑖 = 1,2,…,2𝑘−1) at the (k-1)th stage shall be of length 𝛼= 𝑑(𝐴𝑘(𝑖)) = 12𝑘−1 13𝑘, in which case 𝑑ቀ𝐵𝑘ሺ𝑖ሻቁ= 12𝑘 (1− σ 13𝑗𝑘𝑗=1 ) and, therefore, 𝑑ሺ𝐶ሻ= 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 ሼ𝑥𝑛∗ሽ𝑛=1∞ . Write 𝐴𝑘(𝑖) = [𝑎𝑘ሺ𝑖ሻ,𝑏𝑘(𝑖)] and define a function 𝜑𝑘(𝑖):[0,1] →ℝ which is piecewise linear on 𝐴𝑘(𝑖) and vanishes off 𝐴𝑘(𝑖).

Put ℎ𝑘ሺ𝑡ሻ= 𝜑𝑘ሺ𝑖ሻሺ𝑡ሻ,𝑡 ∈ሾ0,1ሿ,2𝑘−1

𝑖=1 and define

𝑓ሺ𝑡ሻ= ℎ𝑘ሺ𝑡ሻ𝑥𝑛∗ ,𝑡 ∈ሾ0,1ሿ.∞𝑘=1

Claim 1: f is weak*-continuous. This is achieved by showing that the series defining f is uniformly convergent in 𝑋𝜎∗.

Claim 2: f is not Riemann integrable. Here we use the fact that the Cantor set C constructed above has measure equal to 1 2ൗ and then produce a bounded subset B of X and tagged partitions 𝑃1 and 𝑃2of [0, 1] such that 𝑝𝐵൫𝑆ሺ𝑓,𝑃1ሻ− 𝑆ሺ𝑓,𝑃2ሻ൯> 1 2ൗ, where 𝑝𝐵 is the strong*-seminorm on X∗ corresponding to B defined by 𝑝𝐵ሺ𝑓ሻ= ax𝑓(𝑥)ax𝑥∈𝐵𝑠𝑢𝑝 ,𝑓∈X∗. 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 𝑓:[𝑎,𝑏] →𝑋 have a primitive F: 𝐹′ሺ𝑡ሻ= 𝑓ሺ𝑡ሻ,∀𝑡 ∈ሾ𝑎,𝑏ሿ, i.e., 𝑙𝑖𝑚ℎ →0ብ𝐹ሺ𝑡+ ℎሻ− 𝐹(𝑡)ℎ − 𝑓(𝑡)ብ= 0,∀𝑡 ∈ሾ𝑎,𝑏ሿ.

The following problem has a slightly different flavour and is motivated by the idea of ‘decomposition’ of a ‘finite dimensional’ property, a phenomenon which has been treated in a recent work of the author “Around finite dimensionality in functional analysis” (RACSAM, 2013). PROBLEM 2: Describe the existence of a locally convex topology 𝜏 on the dual of a Banach space X such that each 𝑓:ሾ𝑎,𝑏ሿ→𝑋∗ continuous w r t 𝜏 is Riemann integrable if and only if X is a Hilbert space.