Transcript WEAKER FORMS OF CONTINUITY AND VECTOR VALUED …

WEAKER FORMS OF
CONTINUITY AND
VECTOR VALUED RIEMANN
INTEGRATION
M.A.SOFI
DEPARTMENT OF MATHEMATICS
KASHMIR UNIVERSITY, SRINAGAR-190006
INDIA
1.Classical Situation
Given a continuous function 𝑓: 𝑎,𝑏 → ℝ, then
a. f is Riemann integrable.
b. f has a primitive: ∃ 𝐹: 𝑎,𝑏 → ℝ such that F is differentiable on 𝑎,𝑏
and 𝐹′ 𝑡 = 𝑓 𝑡 𝑜𝑛 𝑎,𝑏 .
2.Banach spaces
Let X be a Banach space and 𝑓: 𝑎, 𝑏 → 𝑋 a continuous function. Then
a. f is Riemann integrable.
b. f has a primitive: ∃ 𝐹: 𝑎, 𝑏 → 𝑋 such that F is differentiable on
𝑎, 𝑏 and 𝐹′ 𝑡 = 𝑓 𝑡 𝑜𝑛 𝑎, 𝑏 .
c. If 𝑓 is differentiable on [a, b], then 𝑓 ′ is Henstock integrable and
𝑥 ′
𝑓 𝑥 = 𝑎 𝑓 𝑡 𝑑𝑡, ∀𝑥 ∈ 𝑎, 𝑏 .
3.Quasi Banach spaces
Let X be a quasi Banach space. Then
a. Continuity of 𝑓: 𝑎, 𝑏 → 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 ∗ = 0 , continuity of f implies f has a primitive.
(In particular, this holds for X= 𝐿𝑝 𝑎, 𝑏 , 0 < 𝑝 < 1).
d. (Fernando Albiac) For X such that 𝑋 ∗ is separating, there exists a
continuous function 𝑓: 𝑎, 𝑏 → 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: 𝑎, 𝑏 → 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 RiemannLebesgue (WRL) - property if 𝑓: 𝑎, 𝑏 → 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 Ω, Ʃ ,
the space 𝐿1 (Ω, Ʃ)has (WLP).
(i) Theorem (Wang and Yang): For a given measurable space Ω,Ʃ ,
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: 𝑎, 𝑏 → 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: 𝑎, 𝑏 → 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: 𝑎, 𝑏 → 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 , 𝑡𝑖−1 , 𝑡𝑖 , 1 ≤ 𝑖 ≤ 𝑗} of [a, b] with
𝑚𝑎𝑥
𝑃 = 1≤𝑖≤𝑗
𝑡𝑖 − 𝑡𝑖−1 < 𝛿,
we have
𝑝𝑛 𝑆 𝑓, 𝑃 − 𝑥 < 𝜀,
where, 𝑆 𝑓,𝑃 is the Riemann sum of f corresponding to the tagged
partition P= {𝑠𝑖 , 𝑡𝑖−1 ,𝑡𝑖 ,1 ≤ 𝑖 ≤ 𝑗} of [a, b] where 𝑎 = 𝑡0 < 𝑡1 < ⋯𝑡𝑗 = 𝑏
and 𝑠𝑖 ∈ 𝑡𝑖−1 ,𝑡𝑖 ,1 ≤ 𝑖 ≤ 𝑗. Here, {𝑝𝑚 }∞𝑚=1 denotes a sequence of
seminiorms generating the (Frechet)-topology of X. The (unique) vector x,
to be denoted by
𝑏
𝑓 𝑡 𝑑𝑡,
𝑎
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.
(𝑖)
In the instant case, each of the 2𝑘−1 subintervals 𝐴𝑘 ( 𝑖 =
1,2,… ,2𝑘−1 ) to be knocked out at the kth stage of the construction
(𝑖)
from each of the remaining subintervals 𝐵𝑘 ( 𝑖 = 1,2,… , 2𝑘−1 ) at the
1 1
(𝑖)
(k-1)th stage shall be of length 𝛼 = 𝑑(𝐴𝑘 ) = 𝑘−1 𝑘 , in which case
𝑖
1
𝑑 𝐵𝑘 = 𝑘 (1 −
2
𝑘 1
𝑗=1 3𝑗 ) and, therefore, 𝑑
2
1
𝐶 = .
2
3
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
(𝑖)
𝑖
(𝑖)
sequence by 𝑥𝑛∗ ∞𝑛=1 . Write 𝐴𝑘 = [𝑎𝑘 ,𝑏𝑘 ] and define a function
(𝑖)
(𝑖)
(𝑖)
𝜑𝑘 : [0,1] → ℝ which is piecewise linear on 𝐴𝑘 and vanishes off 𝐴𝑘 .
Put
2𝑘−1
𝑖
ℎ𝑘 𝑡 =
𝜑𝑘 𝑡 , 𝑡 ∈ 0, 1 ,
𝑖=1
and define
∞
ℎ𝑘 𝑡 𝑥𝑛∗ , 𝑡 ∈ 0, 1 .
𝑓 𝑡 =
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 2 and then produce a bounded subset B of X and
tagged partitions 𝑃1 and 𝑃2 of [0, 1] such that
𝑝𝐵 𝑆 𝑓, 𝑃1 − 𝑆 𝑓, 𝑃2
> 1 2,
where 𝑝𝐵 is the strong*-seminorm on X ∗ corresponding to B defined
by
𝑠𝑢𝑝
𝑝𝐵 𝑓 = 𝑥∈𝐵 𝑓(𝑥) , 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′ 𝑡 = 𝑓 𝑡 , ∀𝑡 ∈ 𝑎, 𝑏 ,
i.e.,
𝑙𝑖𝑚 𝐹 𝑡 + ℎ − 𝐹(𝑡)
− 𝑓(𝑡) = 0, ∀𝑡 ∈ 𝑎, 𝑏 .
ℎ→0
ℎ
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: 𝑎,𝑏 → 𝑋 continuous
w r t 𝜏 is Riemann integrable if and only if X is a Hilbert space.