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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Convergence : $ ||u_n(.)||^2_v \rightarrow ||u(.)||^2_v$ in $L^1(]0,T[)$?

if I have a sequence $(u_n)_n$ that converges to $u$ in $L^2(]0,T[,v)$ , then why $ ||u_n(.)||^2_v \rightarrow ||u(.)||^2_v$ in $L^1(]0,T[)$? Remark that $L^2(]0,T[,v)$ is the space of measurable ...
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20 views

Convergence in $L^2(]0,T[,v)$

if I have a sequence $(u_n)_n$ that converges to $u$ in $L^2(]0,T[,v)$ , then why $ ||u_n(.)||^2_v \rightarrow ||u(.)||^2_v$ in $L^1(]0,T[)$? Remark that $L^2(]0,T[,v)$ is the space of measurable ...
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29 views

Relations between Bochner spaces $L^p(0,T;L^q)$

Let $(X,\|\cdot\|_X)$ be a Banach space. The function $u=u(t,x)$ belongs to Bochner space $L^p(0,T;X)$ if the norm $$ \|u\|_{L^p(0,T;X)} = \left(\int_0^T \|u(t,\cdot)\|_X^p \mathrm{d}t\right)^{1/p} \...
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25 views

Does weak convergence in $L^\infty(\mathbb{R},L^p(\mathbb{R}^n))$ imply strong convergence in $L^p_{loc}(\mathbb{R}\times\mathbb{R}^n)$?

Does weak convergence in $L^\infty\left(\mathbb{R},L^p\left(\mathbb{R}^n\right)\right)$ imply strong convergence in $L^p_{loc}\left(\mathbb{R}\times\mathbb{R}^n\right)$ up to a subsequence ?
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28 views

Why do we obtain $C^{1+\alpha,\frac{1+\alpha}{2}}(U\times(0,T))$ regularity?

Let $U$ be a $C^3$ compact manifold in $\mathbb R^3$ and assume $f \in L^2(0,T;H^1(U)) \cap H^1(0,T;H^1(U)^*) \cap W^{2,1}_p(U\times (0,T))$ for all $p \in[1,\infty)$, where $H^1(U)^*$ denotes the ...
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20 views

Example of Non-Linear but Continuous Functionals on Bochner Space

Let $(X,\Sigma,\mu)$ be a finite measure space and $E$ be a separable Banach space (not $0$-dimensional). Then, what are some examples of non-linear functionals on the Bochner-space $L^1_{\mu}(X,E)$ ...
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28 views

weak star convergence and time derivative in Bochner spaces

Assume that for a sequence of vector valued functions $u(\varepsilon):(0,T)\to L^2(\Omega)^3$ we have $\frac{1}{2}(\partial_i u_j(\varepsilon)+\partial_j u_i(\varepsilon))\stackrel{*}{\rightarrow}e_{...
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16 views

A regularity result in Bochner spaces, simmetry of second derivatives (time and space)

For a function $u:(0,T)\to L^2(\Omega)$, assume that we have shown that $\frac{\partial u}{\partial x_i}=\partial_i u \in L^\infty(0,T;L^2(\Omega))$ and also that $\partial_t u \in L^\infty(0,T;L^2(\...
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40 views

How to construct a function in $C_{wk}([0,T];L^2(\Omega))$, but may not in $C([0,T];L^2(\Omega))$?

The question is in the third chapter of this book http://www.ams.org/publications/authors/books/postpub/gsm-192 The notation $C_{wk}([0,T];L^2(\Omega))$ mean: A function f(t):$[0,T]\to L^2(\Omega)$ is ...
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23 views

If $u \in L^2(0,T;L^2(\Omega))$ with $u(t) \in L^\infty(\Omega)$ uniformly, is $u \in L^\infty(0,T;L^\infty(\Omega))$?

Suppose I have a function $u \in L^2(0,T;L^2(\Omega))$. If I know that for almost all $t$, $\lVert{u(t)}\rVert_{L^\infty(\Omega)} \leq C$ for some constant, does it follow that $u \in L^\infty(0,T;L^\...
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36 views

positive part of Banach space-valued Sobolev map

I refer to: W. Arendt, M. Kreuter - Mapping theorems for Sobolev spaces of vector-valued functions (https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/kreuter/adoi-sm8757-4-2017.pdf) ...
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22 views

Convergence of mollifed Bochner integrable functions

I have a question about Spaces involving time from Evans' PDE book. Let $u \in L^2(0,T; H^1_0(U))$, with $u' \in L^2(0,T; H^{-1}(U))$. Could we show that $(u_\epsilon)' \to u'$ in $ L^2(0,T; H^{-...
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30 views

If $u_n \in L^\infty(0,T;L^\infty)$ and $u_n \rightharpoonup^* u$ in $L^\infty((0,T)\times \Omega)$, is $u \in L^\infty(0,T;L^\infty)$?

Let $\Omega$ be a smooth bounded domain. If $u_n \in L^\infty(0,T;L^\infty(\Omega))$ and $u_n \rightharpoonup^* u$ in $L^\infty((0,T)\times \Omega)$, is $u \in L^\infty(0,T;L^\infty(\Omega))$? Note ...
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31 views

$L^p(X; L^q(Y)) \cong L^q_yL^p_x(X \times Y)$?

When I'm studying Strichartz estimates for Schrödinger equations, Bochner spaces $L^p(I;L^q(\mathbb{R}^n))$ appeared. However, I feel that it is technically difficult to handle $L^p(I; L^q(\mathbb{R}^...
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79 views

Proof of Pettis' theorem about strong and weak measurability.

I am studying Bochner integral, reading K.Yosida "Functional Analysis 6th edition." However, I cannot understand the proof of Pettis' theorem, which is in p.131-132. From my perspective, ...
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1answer
43 views

Why the sequence $(E_n)$ exists

Let $(\Omega,\Sigma,\mu)$ be a finite measure space and $(X,\|.\|)$ be a reflexive Banach space. I did not understand why the sequence $(E_n)$ exists. An idea please.
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59 views

Understanding the interplay between Fréchet derivative and ordinary derivative.

Let $Y$ be a Banach space. Consider the real-valued fucnion $\phi \in C^\infty _c (\Omega)$, where $\Omega \subset \mathbb{R}^n$ is open. Let $y\in Y$. Then can we say $\phi y \in C^\infty _c (\Omega; ...
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34 views

show that : $t \mapsto \langle x^{*},f(t)\rangle_{X^{*},X} $ is measurable from $E$ to $\mathbb{R}$

Let $(E,\mathcal {A},\mu)$ be a finite measure space and let $ (X, \|\cdot\|)$ be a reflexive Banach space. Let $\{f_n\}\subset \mathcal{L}_{X}^{1}$ be a sequence with : $$ \sup_{n}\int_{E}\|f_n\|<+...
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80 views

Weak convergence of simple functions implies strong convergence?

For each $n$, let $\{A_j^n\}_j$ be a disjoint partition of the interval $[0,1]$ such that as $n$ increases, the partition gets finer. Suppose that we have $$f_n(t) = \sum_{j=1}^n a_{jn}\chi_{A_j^n}(t)...
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45 views

Uniform bound in $L^\infty(0,T;X)$ implies pointwise a.e. convergence in $Y$ if $X \subset Y$ is compact?

Suppose that $u_n \in L^\infty(0,T;X)$ is a uniformly bounded sequence, so $u_n \to u$ weakly-star to some $u$. and that $X \subset Y$ is a compact embedding of Hilbert spaces. Does it follow that ...
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51 views

Relation between weak limits

Let $H$ be a Hilbert space such that $H\hookrightarrow \hookrightarrow L^2(\mathbb{R}^d)$. Let $\{u_n\}$ be a sequence such that $u_n$ converges strongly to $v$ in $C([0,T]; H')$ and converges weakly ...
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46 views

Bochner Space “basic properties” (Reference Request)

I'm looking for references to what I believe to be "basic properties" of Bochner Space of the type $$L^p(\Omega; X),$$ where $X$ is a Banach space and $\Omega \subset \mathbb{R}^N$ is an opened and ...
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68 views

Compact embedding in Bochner space

Let $V \subset H$ be a compact embedding of Hilbert spaces. Suppose $u_n$ is bounded uniformly in $L^2(0,T;V)$. Does it follow that (for a subsequence), $u_{n_j}(t) \to u(t)$ in $H$ for some $u(t) \in ...
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41 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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51 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 ...
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1answer
65 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\...
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1answer
23 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 ...
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84 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)&\...
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52 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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1answer
28 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 ...
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1answer
70 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 ...
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1answer
50 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
27 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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22 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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73 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
177 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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305 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
112 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
68 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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18 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
29 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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83 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
65 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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1answer
96 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
110 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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29 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
50 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
30 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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206 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
84 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 ...