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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Equivalent Lebesgue integrability

I'm reading section "Integration of $\overline{\mathbb{R}}$-valued functions" on page 103 from Amann's texbook Analysis III. The decomposition of an $\overline{\mathbb{R}}$-valued function ...
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Bochner integral on function spaces

Let $(X,\Sigma,\mu)$ be a measure space and $B$ be a Banach space. A Bochner-measurable function $f: X \rightarrow B$ is Bochner integrable if and only if $$ \int_{X}\|f(x)\|_{B} d \mu(x)<\infty $$ ...
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About the proof of Bochner Theorem.

Theorem 8(Bochner) A strongly measurable function $f$ :[0,T]$\rightarrow$ X is summable if and only if t$\rightarrow$ $\left\|f\right\|$ is Lebesgue summable. the proof of "if" part. Let ${...
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Questions about the Bochner integral

I'v seen a proposition saying that: Let $f: \Omega \rightarrow \mathcal{X}$ be a Bochner integrable function and $\mu$ a $\sigma$-finite measure. Let $\left(A_{n}\right)_{n \in \mathbb{N}}$ be a ...
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Haar integral, invariance and matrices

Wikipedia says: One property of a left Haar measure $\mu$ is that, letting $s$ be an element of $G$, the following is valid: $$\int_G f(sx) d\mu(x) = \int_G f(x) d\mu(x)$$ for any Haar integrable ...
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Crossing out an orthogonal-valued function out of expected value of product

Main part: Let $G$ be a locally compact group with corresponding Haar measure $\mu$. Let $f, g: \Omega \to G$ be arbitrary measurable functions. It would be really nice to have: $$\mathbb{E} [gf] = \...
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Convergence in Bochner space.

Let $\Omega \subset \mathbb{R}^n$ be a bounded set and $\{f_n\}_{n \in \mathbb{N}}$ be a sequence which converges to $f$ in $C([0,T];L^2(\Omega))$. I need to prove that $\int_\Omega f_n^2 \phi \,dx \...
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Proof of an identity in a Lebesgue-Bochner space

Let $u \in L^2(0,T;H^1(\Omega))$ such that $\partial_t u \in L^2(0,T;(H^1(\Omega))^*)$ and define the following function $$ f_\epsilon(s)=\begin{cases}s \;\; \text{if} \hspace{0.4cm} \vert s\vert \leq ...
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Differentiability in time and space in Bochner Spaces

Could anybody please help me check whether the following statement is true? For $T>0$ and $\Omega = (0,1)$, let $f, g \in C^1([0,T];L^2(\Omega))\bigcap C([0,T];H^1(\Omega))$ such that $$\min_{t \in ...
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How to prove this condition for Bochner integrability on a general measure space?

Let $f: \Omega \rightarrow B$ be Bochner-measurable, i.e. the point-wise limit of a sequence of simple (i.e. countably-valued measurable) functions $(s_n)$. I know that if $$ \int_{\Omega} ||f(\omega)|...
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Does boundedness in $L^\infty ( 0,T; L^1 (\Omega))$ imply boundedness in $L^\infty (0,T;BV(\Omega))$?

Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$ and let $T>0$. Assume $\lVert \nabla u \rVert_{L^\infty ( 0,T; L^1 (\Omega))} \leq C_1$. Does this imply $\lVert u \rVert_{L^\infty (0,T;BV(\...
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Are weak-star limits in $L^\infty (0,T;L^2(\Omega))$ and in $L^\infty (\Omega \times \left[ 0,T \right])$ equal?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and let $T>0$. Let $u_n$ be a sequence such that there exists a subsequence (still denoted by $u_n$) \begin{equation} u_n \rightarrow u \...
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Is $C([0,T];C[a,b]) = C([0,T] \times [a,b])$

I came across a doubt yesterday. Is the following equality true?? $$C([0,T];C[a,b]) = C([0,T] \times [a,b])$$ I have a feeling that the answer is no. I know that if $f \in C([0,T];C[a,b])$ then $f \in ...
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Strong convergence and a.e. convergence in Bochner spaces

Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$. Let $u_n \longrightarrow u$ strongly in $L^2(0,T;L^p(\Omega))$ for $1 \leq p < \infty$. It is known that strong convergence in $L^2$ ...
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Does weak convergence in $L^2(0,T;L^2(\Omega))$ imply weak convergence in $L^2(0,T;L^1(\Omega))$?

Let $\Omega$ be an open bounded subset in $\mathbb{R}^n$ Does weak convergence in $L^2(0,T;L^2(\Omega))$ imply weak convergence in $L^2(0,T;L^1(\Omega))$? I tried to prove it the following way: Let $...
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Is there a relation between weak-* convergence in $L^\infty((0,T); L^2(\Omega))$ and strong convergence in $L^2(\Omega)$ uniformly in $t$?

I have problems in understanding a proof. So my question is: Let $u_n$ be a sequence that converges weak-* in $L^\infty((0,T); L^2(\Omega))$ to $u$. Can I imply that $u_n$ converges strongly in $L^2(\...
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A question on Bochner's integral.

Let $(S, \mathcal A, \mu)$ be a measure space and $x : S \longrightarrow X$ be a function to a Banach space $X.$ Then $x$ is Bochner integrable if there exists a sequence of simple measurable ...
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If $S(t)$ is a $C_0$-semigroup, is $S(t-s)f(s)$ Bochner integrable?

Let $X$ be a Banach space and let $S(t)$, $t \geq 0$, be a $C_0$-semigroup on $X$. Assume that $f : [0,+\infty) \rightarrow X$ is Bochner integrable. Is $S(t-s)f(s)$ Bochner integrable on $[0,t]$ and ...
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Almost everywhere convergence of subsequence in Bochner space

Does the familiar theorem that convergence in $L^p$ implies almost everywhere convergence along a subsequence also hold in Bochner spaces? I.e., for a Banach space $B$ does $f_n\to f$ in $L^p(\Omega;B)...
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Measurable and norm has finite integral implies strongly measurable?

In the theory of Bochner integration (taking definitions from Yosida), letting $(E, \mathcal{F},\mu)$ a complete measure space and $X$ a Banach space, there's a simple result (attributed to Bochner) ...
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Measurability of the Barycenter Map

Let $V$ be a Banach space considered with its Borel $\sigma$-algebra $\mathcal{B}(V).$ Let $V^*$the dual of $V.$ A probability measure $\mu$ on $(V, \mathcal{B}(V))$ is said to have a barycenter if ...
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A.e. $L^p$ convergence implies a.e. convergence along sub-sequence

The book on Bochner spaces that I am currently looking at contains the following theorem: Let $\tilde{u}:[0,T] \to L^p(a,b)$ be Bochner measurable for some $1\leq p< \infty$. Define $$u: [0,T] \...
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Does measurability on product space implies strong measurability?

In Giovanni's book, A First Course in Sobolev Spaces, he says Theorem 8.28. Let $I \subset \mathbb{R}$ be an open interval, let $E \subset \mathbb{R}^n$ be a Lebegue measurable set, and let $1 \le p &...
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reflexivity of Bochner space

Let $(\Omega, \mathscr F, \mu)$ be a $\sigma$-finite measure space and $X$ be a Banach space, and assume that $X^*$ has the Radon-Nikodym property with respect to $(\Omega, \mathscr F, \mu)$. I'm ...
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Measurability of Banach space valued function

If $u \in L^2(0,T; L^2(\mathbb{R}^d))$, then can we say $u$ is measurable on $(0,T) \times \mathbb{R}^d$? I tried to use the simple functions, which converge to $u(t)$ with respect to $\Vert \cdot\...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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