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.
0
votes
1answer
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 \...
0
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0answers
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 ...
3
votes
0answers
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);\...
1
vote
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,...
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
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 $...
1
vote
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_{...
1
vote
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 ...
0
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0answers
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 ...
1
vote
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,...
1
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0answers
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$ ...
1
vote
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 ...
2
votes
1answer
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, \...
2
votes
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 ...
1
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0answers
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{...
2
votes
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),...
0
votes
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 ...
1
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0answers
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 $\...
1
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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 ...
1
vote
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 ...
0
votes
0answers
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^...
2
votes
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 ...
1
vote
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 $\Vert x\Vert_Y \leq c\...
0
votes
1answer
51 views
Weak limit in Bochner-Sobolev space
Let "$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
$$u_n(t)\to u_1(t)\qquad\text{weakly in} \qquad V, $$
$$u_n'(t)\to u_2(t)...
4
votes
2answers
134 views
Weak continuity of functions with values in a Banach space
I have problems understanding the proof of the following lemma:
Let $X$, $Y$ be Banach spaces, $X$ reflexive, and assume that $X$ is continuously, densely embeded into $Y$. Let $I \subset \mathbb{R}$ ...
1
vote
0answers
41 views
Show that the integral of a two-parameter function is Hölder continuous
Let
$(\Omega,\mathcal A,\mu)$ be a measure space
$(M,d)$ be a metric space
$\Lambda\subseteq M$
$E$ be a $\mathbb R$-Banach space
$f:\Omega\times\Lambda\to E$ with $$f(\;\cdot\;,x)\in\mathcal L^1(\mu;...
1
vote
0answers
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 ...
1
vote
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?
3
votes
0answers
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 ...
2
votes
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 ...
0
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0answers
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?
2
votes
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) = ...
1
vote
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 $$\left\|\...
0
votes
1answer
138 views
Weak star convergence in $L^\infty(0,T;L^2(\Omega))$ and pointwise weak limits
Let $\Omega$ be a bounded domain of $\mathbb{R}^n$ and $u_n \colon [0,T] \to L^2(\Omega)$ a sequence of functions such that $||u_n(t)||_{L^2(\Omega)} \leq C$ for every $n$ and for every $t \in [0,T]$.
...
2
votes
0answers
64 views
Bochner integral of a function in $L^2$
This might be obvious, but I don't see why...
Let $\lambda \in \mathbb{C}\backslash\mathbb{R}_+$ and $m(x) = \lambda/(\lambda - e^x)$. If $f \in L^2$, then
$$
mf = \int^\infty_{-\infty} \frac{\...
2
votes
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 ...
2
votes
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 ...
0
votes
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)...
1
vote
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/...
2
votes
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 ...
4
votes
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 $...
1
vote
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(\...
3
votes
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 ...
1
vote
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
votes
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(\...
1
vote
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
votes
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}(\...
4
votes
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. ...
1
vote
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
votes
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 ...
