Mathematics Stack Exchange
Mathematics Stack Exchange

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.

275 questions
16
votes
2 answers

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
  • 10,804
15
votes
1 answer

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
  • 941
11
votes
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
  • 131
  • 3
7
votes
2 answers

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
  • 71
6
votes
0 answers

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
  • 781
6
votes
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
  • 716
6
votes
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…
Akira
  • 18,439
6
votes
1 answer

$L^{2}(0,T; L^{2}(\Omega))=L^{2}([0,T]\times\Omega)$?

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
  • 155
6
votes
0 answers

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…
0xbadf00d
  • 14,208
6
votes
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
  • 1,425
6
votes
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
  • 4,737
  • 1
  • 23
  • 43
5
votes
0 answers

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
  • 8,357
5
votes
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
  • 6,351
5
votes
1 answer

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
  • 1,963
5
votes
1 answer

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

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. What is the meaning of $u_n \rightharpoonup^\star u$ in $L^{\infty}(0,T;H^1(\Omega))$? Thank you!
Nicole
  • 51
1
2 3
…
18 19