Mathematics

Questions tagged [bochner-spaces]

For question involving Bochner space, which are generalization $\mathbb L^p$ spaces in the sense that the values of the functions are themselves in function spaces.

273 questions
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Is there a notion of a continuous basis of a Banach space?

If $X$ is a Banach space, then a Hamel basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as a linear combination of elements of $B$ . And a Schauder basis of $X$ is a subset $B$ of $X$ such that every element…
Keshav Srinivasan
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15
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Are continuous functions strongly measurable?

Measure theory is still quite new to me, and I'm a bit confused about the following. Suppose we have a continuous function $f: I \rightarrow X$ , where $I \subset \mathbb{R}$ is a closed interval and $X$ is a Banach space. I can show that $f$ is…
ScroogeMcDuck
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11
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2 answers

Weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$

What does weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$ means? $\Omega$ is open, bounded, has boundary smooth and etc...
user119148
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Exercise 4 chapter 7 Evans book

I want to solve the following problem: Suppose $H$ is a Hilbert space, if $u_{n}\rightharpoonup u$ in $L^{2}(0,T;H)$ and $u'_{n}\rightharpoonup v$ in $L^{2}(0,T;H^{'})$. I want to prove that $v=u'$. Thanks for your help.
Miguel
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Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper . In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $\frac{\partial b(\overline{v}_n)}{\partial t}$ is only…
riem
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6
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2 answers

$L^\infty((0,T)\times\Omega)$ is not equal to $L^\infty(0,T;L^\infty(\Omega))$.

Let $\Omega$ be bounded domain in $\mathbb{R}^n$. We know that $L^\infty((0,T)\times\Omega)$ is not equal to $L^\infty(0,T;L^\infty(\Omega))$. Are there any circumstances in which we can say that one is a subset of the other though? Is it really…
student
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3 answers

Bochner integral: Is $f=g$ $\mu$-a.e. if their integrals are equal on every measurable set?

Let $(X, \mathcal F, \mu)$ be a $\sigma$ -finite complete measure space, and $(E, |\cdot|_E)$ a Banach space. Assume $f,g:X \to E$ are $\mu$ -integrable such that $$ \int_A f \mathrm d \mu = \int_A g \mathrm d \mu \quad \forall A \in \mathcal F. $$…
Akira
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6
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$L^{2}(0,T; L^{2}(\Omega))=L^{2}([0,T]\times\Omega)$?

Suppose I have a function $u\in L^{2}([0,T]\times\Omega)$ for some bounded domain $\Omega$ in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$. I managed to prove that this implies $u\in L^{2}(0,T; L^{2}(\Omega))$. Now I want to show that opposite is not true.…
MajinSaha
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How do theorems like the optional stopping theorem generalize to Bochner integrable processes with values in a separable Banach spaces?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space $(X_t)_{t\ge 0}$ be an $\mathcal F$-adapted and integrable…
0xbadf00d
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6
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3 answers

Integration in Banach spaces

Let $X$ be a Banach space and let $L = \{f:[0,1]\to X\vert\, f \text{ Borel-measurable}, \int_0^1 \Vert f \Vert < + \infty \}$ ($\int$ being the Lebesgue integral.) Now define $$ T:L \to X^{**} \quad \text{by} \quad (Tf)(x^*) = \int_0^1 x^*\circ f…
Zardo
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6
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1 answer

Bochner-Sobolev space definition

I take course in PDE and I'm a little bit puzzled with space $W^{1,p}(0,T,X)$. Evans defines in §5.9.2 $W^{1,p}(0,T,X)$ as follows $$ W^{1,p}(0,T,X) = \{ u \in L^p(0,T,X): u'\in L^p(0,T,X)\} $$ but solution $u$ to the parabolic system is defined in…
tom
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5
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How to prove the isometry $(L^p(S;X))^*=L^q(S;X^*)$ when $S$ is an atomic measure space?

I meet a problem in Analysis in Banach Space (Volume I) . At page 39 the authors want to prove the following result: Proposition 1.3.3. Let $1<p<\infty$ and $1/p+1/q=1$ , and let $(S,\mathscr{A},\mu)$ be an atomic measure space. Then for any Banach…
ununhappy
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Property of vector-valued measure

Let $B$ be a Banach space, let $(X,\mathcal{A})$ be a measurable space, and let $\mu:\mathcal{A}\to B$ be a vector-valued measure of bounded variation. In general, if $B$ doesn't have the Radon-Nikodym property, it is not true that we can express…
geodude
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5
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1 answer

Let $E$ be a Banach space, $p \in (1, \infty)$, and $L_p := L_{p}(X, \mu, E)$. Is $(L_p)^* \cong L_{p'}$ where $\frac{1}{p} + \frac{1}{p'} = 1$?

Let $(X, \Sigma, \mu)$ be a $\sigma$ -finite measure space and $(E, |\cdot|)$ a Banach space. Here we use Bochner integral . The space $L_p := L_{p}(X, \mu, E)$ consists of (equivalence classes of) all Bochner measurable functions $f$ with values in…
Analyst
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If $S(t)$ is a $C_0$-semigroup, is $S(t-s)f(s)$ Bochner integrable?

Let $X$ be a Banach space and let $S(t)$ , $t \geq 0$ , be a $C_0$ -semigroup on $X$ . Assume that $f : [0,+\infty) \rightarrow X$ is Bochner integrable. Is $S(t-s)f(s)$ Bochner integrable on $[0,t]$ and does $t \mapsto \int_0^t S(t-s)f(s)ds \in…
Desura
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