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.
151
questions
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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),\...
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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
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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.
$$
f(x)=\int_{\mathcal{H}} e^{i\langle x,...
1
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0answers
27 views
Compactness in Bochner-spaces
Consider a sequence of functions $f_n: [0,T]\times [0,T] \times \Omega \rightarrow \mathbb{R}^3 $, $\Omega \subset \mathbb{R}^3$, such that
\begin{align}
t &\rightarrow f_n(t,s,x) \ \ \ \text{is ...
1
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1answer
35 views
Riesz representation theorem for Bochner integral
Suppose $X$ and $W$ are Hilbert space. Let $F:X\to W$ be a vector-valued function which is Bochner integrable. Define functional $L:W\to\mathbb{C}$ as follows
$$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 $$
\begin{aligned}
L^2_{\nu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D)&\...
1
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0answers
46 views
Differentiability of inner product involving Bochner functions
Suppose we are given $f\in L^2(0, T; H^1)$ and that there exists $\xi\in L^2(0, T; H^1)$ such that, for any $g\in L^2(0,T; H^1)$ with $\dot{g}\in L^2(0, T; H^1)$, it holds that
$$t\mapsto (\nabla f(t)...
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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
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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
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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)...
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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
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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
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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
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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
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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
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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
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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 ...
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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
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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,...
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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 ...
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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),...
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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 ...
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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
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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
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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
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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
$$u_n(t)\to u_1(t)\qquad\text{weakly in} \qquad V, $$
$$u_n'(t)\to u_2(t)...
4
votes
2answers
170 views
Weak continuity of functions with values in a Banach space
I have problems understanding the proof of the following lemma:
Let $X$, $Y$ be Banach spaces, $X$ reflexive, and assume that $X$ is continuously, densely embeded into $Y$. Let $I \subset \mathbb{R}$ ...
1
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0answers
43 views
Show that the integral of a two-parameter function is Hölder continuous
Let
$(\Omega,\mathcal A,\mu)$ be a measure space
$(M,d)$ be a metric space
$\Lambda\subseteq M$
$E$ be a $\mathbb R$-Banach space
$f:\Omega\times\Lambda\to E$ with $$f(\;\cdot\;,x)\in\mathcal L^1(\mu;...
1
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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
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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
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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/...
