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.

Filter by
Sorted by
Tagged with
0
votes
1answer
24 views

Bochner integral in sequence spaces

Let $X$ be a (real) sequence Banach space, such as $c_{0},\ell_{p}$, etc. and $f:[0,1]\longrightarrow X$ continuous. Then, it seems "quite intuitive" that if $f(t):=(f_{1}(t),f_{2}(t),\ldots,f_{n}(t),\...
0
votes
0answers
5 views

Relating $L^{2m}(\Omega;L^2(\mathcal{D}))$-norm with $L^2(\Omega;L^2(\mathcal{D})))$-norm.

Let $(\Omega,\Sigma,\mathbb{P})$ be some probability triple. Let $\mathcal{D} \subset \mathbb{R}^n$ be some nice, closed bounded domain (it can be whatever, really, it's just nice and easy to deal ...
1
vote
0answers
29 views

Fourier transform of measure and Bochner's integral [closed]

Let $\mathcal{H}$ be a real separable Hilbert space and $f:\mathcal{H} \to V$ ($V$ is a topological vector space over $C$) a Fourier transform of measure, i.e. $$ f(x)=\int_{\mathcal{H}} e^{i\langle x,...
1
vote
0answers
27 views

Compactness in Bochner-spaces

Consider a sequence of functions $f_n: [0,T]\times [0,T] \times \Omega \rightarrow \mathbb{R}^3 $, $\Omega \subset \mathbb{R}^3$, such that \begin{align} t &\rightarrow f_n(t,s,x) \ \ \ \text{is ...
1
vote
1answer
35 views

Riesz representation theorem for Bochner integral

Suppose $X$ and $W$ are Hilbert space. Let $F:X\to W$ be a vector-valued function which is Bochner integrable. Define functional $L:W\to\mathbb{C}$ as follows $$L(f):=\int_X\langle f,F(x)\rangle_W d\...
0
votes
1answer
22 views

Long time stability in nested Bochner spaces

Let $V\subset H$ be densely, compactly and continuously embedded, which, among others, means that $$\|x\|_H \leq \|x\|_V\quad (*)$$ for all $x\in V $. I have the following stability estimates for a ...
3
votes
0answers
48 views

Integral of a lsc function is lsc?

Let $f:\mathbb{R}^d\times \mathbb{R}^D\rightarrow \mathbb{R}$ be a lower semicontinuous function. Then is it true that $$ \begin{aligned} L^2_{\nu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D)&\...
1
vote
0answers
46 views

Differentiability of inner product involving Bochner functions

Suppose we are given $f\in L^2(0, T; H^1)$ and that there exists $\xi\in L^2(0, T; H^1)$ such that, for any $g\in L^2(0,T; H^1)$ with $\dot{g}\in L^2(0, T; H^1)$, it holds that $$t\mapsto (\nabla f(t)...
0
votes
0answers
17 views

When do the Strong and Weak Integrals Match?

Let $(\Omega,\Sigma,\mu)$ be a finite-measure space and $X$ be a Banach space over $\mathbb{R}$. When do the Bochner and the Pettis Integral (with values in $X$) agree? Must $X$ be finite-...
0
votes
1answer
23 views

If $\phi$ is $\mu$-Bochner integrable, is it $\nu$-Bochner integrable for $\nu \ll \mu$?

Let $(\Omega, \mathcal{B}(\Omega), \mu)$ be a probability space, $B$ be a separable Banach space. Let $\phi : \Omega \to B$ be a bounded (say by $C$) measurable function, which is Bochner integrable ...
3
votes
1answer
57 views

On DiPerna-Lions compactness' arguments.

In DiPerna Lions, Ordinary differential equations, transport theory and Sobolev spaces (1989), the authors used topological arguments that remains obscure to me. Page 515 the authors used an ...
1
vote
1answer
42 views

Definition de $L^1((0,T);W^{1,\infty}(\mathbb R))$.

I have some trouble about the definition of the space $L^1((0,T);W^{1,\infty}(\mathbb R)$, on one side the formal definition given by Bochner consists in the set of measurable functions $u$ from $(0,T)...
0
votes
1answer
21 views

Measurability of $\nabla \cdot (a \cdot \nabla u)$ implies measurability of $u$?

Suppose I know that $\nabla \cdot (a(t)\nabla u(t))$ is such that $\nabla \cdot (a\nabla u) \in L^2(0,T;L^2(\Omega))$ on some bounded domain $\Omega$. Here $a\colon [0,T] \to \Omega$ is such that $a \...
0
votes
0answers
17 views

Automorphism induced automorphism of Lp spaces

Let $(\mathbb{R}^d,\mathbb{B}(\mathbb{R}^d),\mu)$ where $\mu$ is a $\sigma$-finite Radon measure. If $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ is a homeomorphism, then does $\Phi$ induce a ...
3
votes
0answers
70 views

Post-Composition By Diffeomorphism And Integrability

Let $d,D$ be positive integers, $p \in [1,\infty)$, $U$ an (non-empty) open subset of $\mathbb{R}^D$, and suppose that $f$ is: In the Bochner-Lebesgue space $L^p_{\mu}(\mathcal{B}(\mathbb{R}^d);\...
1
vote
2answers
99 views

Weak star convergent sequence in $L^\infty(0,T; L^2(\Omega))$

Given a sequence $(u_n)_{n\in \mathbb{N}} \subseteq L^\infty (0,T; L^2(\Omega)) \cap H^1(0,T;L^2(\Omega))$ with \begin{align*} u_n \overset{\ast}{\rightharpoonup} u \,\, \text{ in } \,\, L^\infty (0,...
0
votes
0answers
12 views

Representation of vector-integral

Suppose that $H$ is a separable Hilbert space, $(X,\Sigma,\mu)$ be a finite measure space and let $L^2(\Sigma,H)$ denote the set of Borel measurable functions from $X$ to $H$ satisfying $$ \int_{x \in ...
8
votes
0answers
158 views

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 $...
1
vote
1answer
44 views

prove a function is $L^{1}$

If we have a function $F:[0,T] \to H^{1}_{0}(\Omega)\times L^{2}(\Omega)$, how exactly would we show that $F\in L^{1}([0,T],H^{1}_{0}(\Omega)\times L^{2}(\Omega))$? Is it enough to prove that $\int_{...
1
vote
1answer
52 views

Extension of weak derivatives in Bochner spaces

I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans: He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In ...
0
votes
0answers
15 views

Superposition of elementary Dirac measures: looking for reference or explanation

Reading a paper concerning transport of measures, I came accross the following sentence: "...the fundamental fact that a generic measure $\mu$ can be written as the superposition of elementary Dirac ...
1
vote
1answer
27 views

How do I interpret the boundedness in this space?

As I was reading the article Non linear elliptic and parabolic equations involving measure data, I came across with the following: the sequence $\{f_n \}$ is bounded in the space $L^1(0,T;W^{-1,...
1
vote
0answers
37 views

Question on Bochner spaces: Convergence of the weak time derivative

Let $U$ be a $C^2-$compact manifold and consider two non negative sequences $f_n,g_n$ such that $f_n \overset * \to f$ weakly in $L^{\infty}(0,T;\mathcal M_{*}(U))$ $g_n \overset * \to g$ ...
1
vote
1answer
26 views

Question on Bochner spaces: $L^1(0,T,L^1(\Gamma))$ bound and convergence of a sequence

Let $\Gamma$ be a compact $C^2$ manifold and suppose that $f_n$ is a non-negative sequence of functions such that: $(*)\forall t\in (0,T)$ it holds: $\int_{0}^t \int_{\Gamma} f_n \le C_1+C_2t$ for ...
2
votes
1answer
48 views

Question on vector valued distributions: weak time derivative calculation

Although I read all the similar posts here, I still can't find the answer in my question so I 'll try to pose it as clear as I can. DEFINITION $1$: Let $f\in L^1_{loc}(I, X)$. Then $\;\;\langle T_f, \...
2
votes
1answer
70 views

Convergence in Lebesgue-Bochner Space $L^{\infty}(0,T,L^1(\Gamma))$

Let $\Gamma$ be a compact $C^2$ manifold and suppose that $f_n$ is a non negative sequence of functions such that ${\vert \vert f_n \vert \vert}_{L^{\infty}(0,T,L^1(\Gamma))} \le C$ I am interested ...
1
vote
0answers
23 views

evolution equation satisfied by projection

Consider the linear parabolic problem (as in Evans) $$\begin{cases} \partial_t u+Lu=f\ \: \mbox{in} \:\Omega \times (0,T]\\ u=0\ \: \mbox{on} \:\partial \Omega \times (0,T]\\ u(\cdot,0)=g \ \: \mbox{...
2
votes
1answer
36 views

Relation between bochner space $L^1(I,X)$ and $C(I,X)$

I am new in Bochner spaces and I have the following problem. I am not able to prove, that if $u \in L^1((0,1),X),u' \in L^1((0,1),X) $ then $u \in C([0,1],X)$, where $X$ is Banach space and $L^1((0,1),...
0
votes
1answer
27 views

Does a continuous function embed a separable space into a separable closed subspace?

Suppose that $X$ is a topological space and $Y$ is a normed vector space, and $f:X\rightarrow Y$ is continuous. In general we know that if $X$ is separable, then the image $f[X]$ will be separable ...
1
vote
0answers
102 views

Role of Pettis measurability theorem in the construction of the Bochner integral

I am reading a brief introduction to Bochner integral, and before introducing it, it is proved the Pettis measurability theorem, which says that a function $f:A \to B$ (with $A$ a measure space with $\...
1
vote
1answer
70 views

The restriction of a variable in a Sobolev function is a.e. Sobolev

Let $I,J$ be bounded open intervals in $\mathbb{R}$. How to show that $W^{1,1}(I\times J) $ is embedded in the Bochner space $L^1(I, W^{1,1}(J))$? Is there a standard reference where this is ...
1
vote
1answer
169 views

Absolutely continuous and differentiable almost everywhere

I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it: "A strongly absolutely continuous function which is differentiable almost everywhere is ...
2
votes
1answer
85 views

Integration in normed vector spaces

Given an interval $I\subset \mathbb{R}$ and a normed vector space $X$, I want to know if I am able to define the Lebesgue space $L^p(I;X)$ of all $p$-integrable functions $f:I\to X$. I know that this ...
1
vote
1answer
81 views

Boundedness of an operator on a Bochner spaces

Given two Banach spaces $(X,\Vert \cdot\Vert_X)$ and $(Y,\Vert \cdot\Vert_Y)$ such that $X\subset Y$ with continuous embedding (i.e. there exits a constant $c>0$ such that $\Vert x\Vert_Y \leq c\...
0
votes
1answer
57 views

Weak limit in Bochner-Sobolev space

Let "$V\subset H\subset V^*$" be an evolution triple. Let $(u_n)$ be a sequence of elements from $W^{1,2}(0,T;V,H)$ such that $$u_n(t)\to u_1(t)\qquad\text{weakly in} \qquad V, $$ $$u_n'(t)\to u_2(t)...
4
votes
2answers
170 views

Weak continuity of functions with values in a Banach space

I have problems understanding the proof of the following lemma: Let $X$, $Y$ be Banach spaces, $X$ reflexive, and assume that $X$ is continuously, densely embeded into $Y$. Let $I \subset \mathbb{R}$ ...
1
vote
0answers
43 views

Show that the integral of a two-parameter function is Hölder continuous

Let $(\Omega,\mathcal A,\mu)$ be a measure space $(M,d)$ be a metric space $\Lambda\subseteq M$ $E$ be a $\mathbb R$-Banach space $f:\Omega\times\Lambda\to E$ with $$f(\;\cdot\;,x)\in\mathcal L^1(\mu;...
1
vote
0answers
56 views

Density of $C_c^\infty((0,T)\times R^d)$ in $L^2(0,T;H^1)$

I have been dealing with the following problem I can't find an answer to: Let $H^1$ be the usual Sobolev space and $L^2(0,T;H^1)$ the Bochner space of square integrable functions $[0,T]\to H^1$. I ...
1
vote
1answer
96 views

Compute the dual of the $L^1$ space of $L^1$-valued functions (Lebesgue-Bochner space)

What is the dual of the space $X=L^1(\mathbb R_{-}, Y)$, where $Y=L^1(\mathbb R)$? How could the paring between an element of the dual and an element of $X$ be represented?
3
votes
0answers
77 views

Sequential Banach-Alaoglu theorem for a Bochner space

I have a question concerning the sequential weak-$*$ compactness in a Bochner space. Let $H$ be some non-separable Hilbert space. Consider the Bochner space $L^{\infty}(0,T;H)$. It is known that this ...
2
votes
1answer
291 views

Arzela-Ascoli-type embedding: Is $H^1(0,T;X)$ compactly embedded in $C([0,T];X)$?

This is a variant of the question Compact Embedding of $W^{1,2}(0,T;ℝ^d)$ in $C(0,T;ℝ^d)$ where we had $X=\mathbb{R}^d$. Let now $X$ be some Banach space. Question: Is $H^1(0,T;X)$ compactly ...
0
votes
0answers
28 views

Does it make sense to define strongly measurable functions on a Polish space instead of a Banch space?

Does it make sense to define strongly measurable functions on a Polish space instead of a on Banach space? Is there any book that deals with this topic?
2
votes
1answer
40 views

Density of simple functions in Bochner space; approximation from below

Let $f \in L^2(0,T;V)$ where $V=H^1_0(\Omega)$ or $V=L^2(\Omega)$. Suppose that $|f| \leq M$ for some constant $M$ almost everywhere. Is it possible to find a sequence of simple functions $f_n(t) = ...
1
vote
2answers
51 views

If $\mu$ is a finite measure and $ν$ is a vector measure with $|\nu|\le C\mu$, are we able to show $\left|\frac{{\rm d}ν}{{\rm d}\mu}\right|\le C$?

Let $(\Omega,\mathcal A,\mu)$ be a finite measure space $E$ be a $\mathbb R$-Banach space with the Radon-Nikodým property $\nu$ be a $E$-valued vector measure on $(\Omega,\mathcal A)$ with $$\left\|\...
0
votes
1answer
183 views

Weak star convergence in $L^\infty(0,T;L^2(\Omega))$ and pointwise weak limits

Let $\Omega$ be a bounded domain of $\mathbb{R}^n$ and $u_n \colon [0,T] \to L^2(\Omega)$ a sequence of functions such that $||u_n(t)||_{L^2(\Omega)} \leq C$ for every $n$ and for every $t \in [0,T]$. ...
2
votes
0answers
68 views

Bochner integral of a function in $L^2$

This might be obvious, but I don't see why... Let $\lambda \in \mathbb{C}\backslash\mathbb{R}_+$ and $m(x) = \lambda/(\lambda - e^x)$. If $f \in L^2$, then $$ mf = \int^\infty_{-\infty} \frac{\...
2
votes
1answer
55 views

How to prove that, $T\left(\int_a^b f(t)dt\right) =\int_a^b T\circ f(t)dt $

Let $T:E\to F$ be a linear and continuous function between the Banach spaces $E$ and $F$. We consider $f:\to [a,b]\to E$ (with $a<b $) be an integrable function in the sense of Bochner How do I ...
2
votes
0answers
275 views

Product rule and integration by parts in Bochner Sobolev space $W^{1,1}(U)$

Assume $v(t)\in H_0^1(U)$ for all $0\leq t\leq T$. The weak derivative $v'$ of $v$ is defined by \begin{equation} \int_0^T \phi'(t)v(t)\,\mathrm{d}t=\int_0^T\phi(t)v'(t)\,\mathrm{d}t\quad\text{for all ...
0
votes
1answer
38 views

Maximal monotone operators - Zeidler's book - question about the proof.

I have a problem with understanding the proof of the proposition 32.10 which I enclose below (E. Zeidler II B). I cannot understand why Zeidler gets from (28) $$\int_{0}^{T}\varphi'(t)v(t)+\varphi(t)...
1
vote
1answer
196 views

Existence of time derivative in Bochner space $L^2(0,T;H_0^1(U))$ (parabolic pde)

Evans defines a function $u=[u(t)](x):=u(t,x)$, \begin{equation} u\in L^2(0,T;H_0^1(U))\quad\text{with}\quad u'\in L^2(0,T;H^{-1}(U)) \end{equation} as a weak solution of a certain parabolic initial/...