Questions tagged [banach-spaces]
A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
1,733 questions
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votes
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Lower bounds for $\|f*g\|_1$ with mean-zero Lipschitz functions on $[0,1]$
Let $f,g \in L^{1}([0,1])$ satisfy
$$
\|f\|_{1}=\|g\|_{1}=1, \qquad \int_{0}^{1} f(x)\,dx=\int_{0}^{1} g(x)\,dx=0,
$$
and assume
$$
f \in \mathrm{Lip}_{L_f}, \qquad g \in \mathrm{Lip}_{L_g}.
$$
...
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views
A question on monotone bases
Let $(X_{n})_{n=1}^\infty$ be a sequence of Banach spaces and let
$$X:=\{(x_{n})_{n=1}^\infty: x_{n}\in X_{n}(n\in \mathbb{N}), \sum_{n=1}^{\infty}\|x_{n}\|_{X_{n}}<\infty\}.$$
Question. Does $X$ ...
- 3,467
4
votes
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answers
142
views
Strict convexity, continuous modulus, and Kadets-Klee property
Let $(X,\|\cdot\|)$ be a Banach space. Assume that $X$ is strictly convex and that its modulus of convexity
$$
\delta_X(\varepsilon)=\inf\left\{1-\left\|\frac{x+y}{2}\right\| \colon \|x\|=\|y\|=1,\ \|...
- 341
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votes
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Condition for dual of operator systems to be isomorphic
Assume $S$ and $T$ are unital, separable operator systems, and assume further that we have an affine homeomorphism between the state space of $S^*$ and the state space of $T^*$. Does that mean that $S^...
- 277
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1
answer
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views
Asymptotically isometric copy of $\ell_1$ and its perturbation
Let $X$ be a Banach space.
Let $(x_n)_n$ be a sequence in $X$ which is asymptotically isometric copy of $\ell_1$, i.e., there exists a null sequence $(\varepsilon_n)_n$ such that
$$\sum_{n\ge 1}^\...
- 685
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views
A question on the Banach space property of a rearrangement invariant function space
Consider a measure space $(S,\mu)$ and assume that $\mu(S)=1$. We consider the quantile function (or nonincreasing rearrangement) of a real valued function $f:S\to\mathbb{R}$ as the function
\begin{...
- 169
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views
Inner–outer factorization from BK-type modulus for Fredholm determinants on a vertical line
Let $${L_s}_{s\in\mathbb C}$$ be a holomorphic family of trace/nuclear-class operators on a Banach (or Hilbert) space for $\Re s\in[\sigma_0,\sigma_1]$, with analytic Fredholm determinant
$$
\Xi(s) := ...
1
vote
1
answer
90
views
$\beta$-complete bases
In 1972, N. J. Kalton introduced the concept of $\beta$-complete bases to characterize the weak sequential completeness of a Banach space with a basis.
Recall that a basis $(x_{n})_{n=1}^{\infty}$ for ...
- 3,467
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votes
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answers
192
views
On norms such that $(x_n)$ weakly convergent to $x$ and $\|x_n\|\to \|x\|$ imply $\|x_n-x\|\to 0$
Let us say that a norm $\|\cdot\|$ on $X$ is nice if each sequence $(x_n)$ weakly convergent to $x$ with $\|x_n\|\to \|x\|$ is norm convergent.
Using $\|x\|^2=\langle x,x\rangle$, it is easy to check ...
- 4,813
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90
views
A certain subalgebra of the commutant
Suppose $X$ is a (separable) Banach space and $T\in\mathcal{B}(X)$ a bounded operator. Let $\mathcal{A}$ be the unital algebra generated by the resolvent operators
$\{(\lambda I- T)^{-1}: \lambda\in \...
- 1,361
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1
answer
128
views
A question on the summing basis of $c_{0}$
Let $(s_{n})_{n=1}^\infty$ be the summing basis of $X=c_{0}$, where $s_{n}=\sum\limits_{i=1}^{n}e_{i}$ for each $n$ and $(e_{n})_{n=1}^\infty$ is the unit vector basis of $c_{0}$. Let $(a_{n})_{n=1}^\...
- 3,467
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What is the dual of the ba space over an open set?
Let $U \subseteq \mathbb R^n$ be an open set.
The Banach space $ba(U)$ is the space of bounded finitely additive signed measures on the Borel $\sigma$-algebra of $U$. Its norm is the variation.
$ba(U)$...
- 5,525
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votes
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answer
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views
Balls which are bounded weak neighborhoods of $0$
Let $E$ be a Banach space. Let $B$ be the closed unit ball of $E$ endowed with the restriction of the weak topology of $E$. For $e\in E$, $r\in\mathbb{R}$ let $B(e,r)$ be the closed ball of radius $r$ ...
- 5,663
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42
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Norm convergence and weak$^{*}$-cluster points of the series in Banach spaces with bases
Let $(x_{n})_{n=1}^\infty$ be a basis for a Banach space $X$. Assume that $(a_{n})_{n=1}^\infty$ is a sequence of scalars such that $\sup\limits_{n}\|\sum\limits_{i=1}^{n}a_{i}x_{i}\|<\infty$. We ...
- 3,467
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Functional-analytic and Banach space approach to spaces of finite signed measures on $\mathbb{R}$ and $\mathbb{R}^d$
I am looking for good references (survey, monograph, or paper with a solid background section) on the Banach space / functional analytic structure of spaces of finite signed measures $\mathcal{M}(X)$, ...