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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.

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19 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 \...
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14 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 ...
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65 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);\...
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2answers
56 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,...
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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 ...
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132 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 $...
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1answer
37 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_{...
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1answer
38 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 ...
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14 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 ...
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1answer
26 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,...
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27 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 $L^{\infty}(0,T;\mathcal M_{*}(U))$ $g_n \overset * \to g$ ...
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1answer
20 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 ...
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31 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, \...
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1answer
51 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 ...
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19 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{...
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1answer
35 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),...
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1answer
26 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 ...
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50 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 $\...
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1answer
55 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 ...
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1answer
105 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 ...
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54 views

Dual pairing Bochner spaces/ negative Sobolev spaces

Assuming $f \in L^2((0,T)\times \Omega)$ and $g \in L^2(0,T;W^{1,2}(\Omega))$, is the following correct? \begin{equation} \int_0^T\int_\Omega fgdxdt = <f,g>_{L^2(0,T;W^{-1,2}(\Omega)) \times L^...
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1answer
75 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 ...
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1answer
72 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\...
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1answer
51 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)...
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2answers
134 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 $I \subset \mathbb{R}$ ...
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0answers
41 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;...
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46 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 ...
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1answer
78 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?
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61 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 ...
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1answer
194 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 ...
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24 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?
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1answer
32 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) = ...
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2answers
50 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\|\...
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1answer
138 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 $t \in [0,T]$. ...
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0answers
64 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{\...
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1answer
52 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 ...
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0answers
230 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 ...
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1answer
34 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)...
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1answer
174 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/...
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0answers
250 views

Dual of an operator

A linear operator $A$ is defined from $L^2(0,T)$ to $C([0,T);\mathbb{H})$ where $\mathbb{H}$ is a Hilbert space. At some point, dual of this operator is needed; a detour to calculate the dual, and ...
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1answer
106 views

Measurability of functions with values in Banach spaces

The function $f:M\subset \mathbb{R}^n\to Y$ with values in Banach space is called measurable iff the following hold 1) The domain is measurable 2) There exists a sequence $(f_j)$ of step functions $...
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1answer
63 views

If $H$ is a Hilbert space with orthonormal basis $(e_n)$, then $X:\Omega\to H$ is integrable iff $\langle X,e_n\rangle$ is integrable for all $n$

Let $H$ be a separable $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $(\Omega,\mathcal A,\mu)$ be a finite measure space $X:\Omega\to H$ If $X\in\mathcal L^1(\...
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0answers
73 views

Can someone help me obtain these a-priori estimates?

I am trying to obtain some estimates on a PDE where the unknown is: $u(t) : \mathbb{R}^n \rightarrow \mathbb{R}^n, t \in [0,T]$. The variational formulation is given below, where the test function ...
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0answers
45 views

If a process $Y=\{Y_{t}:t\in[0,T]\}$ is progressive, then $\{\int_{0}^{t}Y(s)\text{ d}s\}$ is *surely* continuous

I am reading a textbook and I am stuck proving a seemingly trivial point (but I believe I am not the only one that was pondering this question and it shows similarity to this question). Let me sketch ...
4
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0answers
178 views

Positive part of functions from Sobolev space involving time

Assume that $u \in W^{1,2}(0,T; W_0^{1,2}(\Omega))$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ and $T <+\infty$. Let $A \subset \Omega \times (0, T)$ be a Lipschitz domain such that $u(\...
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1answer
136 views

Gelfand triples and embedding of vector-valued spaces

Suppose $u\in L^2(0,T; V)$ and $u_t\in L^2(0,T;V')$, where $V\subset H\subset V'$ is a Gelfand (or Hilbert) triple (the embeddings are dense and continuous and the spaces are all Hilbert). For ...
3
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1answer
62 views

Show: $W^{-1,2}\left((0,T);W^{0,2}(\Omega)\right)\subset W^{-1,\infty}\left((0,T);W^{-1,2}(\Omega)\right)$

In the paper 'On the Existence of the Pressure for Solutions of the Variational Navier-Stokes Equations' by J. Simon he first defines on page 229 that $\partial_t u \in W^{-1,2}\left((0,T);W^{0,2}(\...
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1answer
96 views

About the dual space of $V=\{u\in H_0^1(\Omega): \text{div}u=0 \}$ and its relations to $H^{-1}(\Omega)$.

I read about some things about $V=\{u\in H_0^1(\Omega): \text{div}u=0 \}$ and its dual space and I began to mix some of these things together. As a result: irritation. I hope you can help me out. ...
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1answer
157 views

Reference for Bochner space.

Are there any books that has a nice introduction to Bochner space including its properties and proofs? Not Evans PDEs. One of my friend recommended me this: https://books.google.co.uk/books?id=...
2
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1answer
56 views

Is $L^1(0,T;V') \subset H^{-1}(0,T;H)$ for $V \subset H \subset V'$?

My question is about a passage in the paper 'On the existence of the pressure for solutions of the variational Navier–Stokes equations' by J.Simon. (It can be found easily via Google). It says the ...