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Results tagged with banach-spaces
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user 99469
A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
0
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107
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If $H \in L_{q} (\mu, X^*)$ such that $\int \langle H, f \rangle \mathrm d \mu = 0$ for all ...
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. Here we use the Bochner integral.
Theorem 1 Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space, $1 \ …
- 825
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277
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Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl.
THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $X$ be a Banach space. …
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1
vote
Accepted
Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?
Below is my formalization of @Nik's hints to finish the proof.
Let's prove that
$$
\sum_{m=1}^M \|H_m\|^q_{L_{p}(\mu_m, X)^*} \le \|H\|^q_{L_{p}(\mu, X)^*} \quad \forall M \in \mathbb N^*.
$$
Let $\O …
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answers
78
views
Does this variant coincide with the usual Hölder space?
$\newcommand{\NN}{\mathbb N} \newcommand{\RR}{\mathbb R}$
Let $\alpha \in (0, 1]$ and $d, j \in \NN^*$.
The usual Hölder space $C^{j, \alpha} := C^{j,\alpha} (\RR^d; \RR)$ is defined as the space of …
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1
answer
157
views
For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_...
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the …
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1
vote
1
answer
57
views
Are there some conditions on a metric space $X$ such that these two types of weak converge o...
Let
$X$ be a metric space,
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$, and
$\mathcal C_b(X)$ be the space of real-valued bounded continuous functions on $X$.
Then $\mathcal …
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1
vote
1
answer
89
views
Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence?
Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Let $p \in [1, \infty)$ and $q \in (1, \infty]$ such that $p^{-1}+q^{ …
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1
vote
1
answer
229
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of Lebe …
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2
votes
1
answer
153
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Are these two norms on localized versions of $L^p_q$ equivalent?
$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$.
Let $E$ be the space of all real-valued …
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1
vote
1
answer
167
views
Rate of convergence of mollified functions in $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a probabi …
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0
answers
64
views
Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Doe...
Let
$X$ be a metric space,
$(E, |\cdot|)$ a Banach space
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal C_b(X)$ be the space of real-valued bounded continuous functio …
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0
votes
1
answer
95
views
Convergence of mollified functions in weighted $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a probabi …
- 825