# 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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### 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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### 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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### 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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### 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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### 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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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 $$L(f):=\int_X\langle f,F(x)\rangle_W d\... 0 votes 1 answer 23 views ### Long time stability in nested Bochner spaces Let V\subset H be densely, compactly and continuously embedded, which, among others, means that$$\|x\|_H \leq \|x\|_V\quad (*)$$for all x\in V . I have the following stability estimates for a ... 2 votes 0 answers 122 views ### Integral of a lsc function is lsc? Let f:\mathbb{R}^d\times \mathbb{R}^D\rightarrow \mathbb{R} be a lower semicontinuous function. Then is it true that$$ \begin{aligned} L^2_{\nu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D)&\... 1 vote
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)...
### 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 ...