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.
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Wildly discontinuous linear functionals
Let $X$ be a Banach space, $H\subseteq X$ be a dense hyperplane, and $f$ be a continuous linear functional defined on $H$. Then $f$ is uniformly continuous and hence it admits a unique continuous ...
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Definition question: asymptotic-$\ell_{p}$ versus coordinate-free asymptotic-$\ell_{p}$
Let $(e_{j})_{j=1}^{\infty}$ be a basis for the Banach space $X$. If there exist constants $\zeta_{1},\zeta_{2}>0$ such that for all $N\in\mathbb{N}$,
\begin{equation*} \zeta_{1}\left(\sum_{i=1}^{N}...
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Integration on quasi-Banach spaces and Schatten ideals
Let $[a,b]$ be an interval and $X$ a Banach space (for starters). We know that continuous functions $f:[a,b]\to X$ are Riemann integrable. Suppose now that $X$ is a quasi-Banach space, that is, its ...
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Example when Kantorovich condition would not hold
Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator
$$
(T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+.
$$
Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...
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A characterization of reflexivity of Banach spaces via convex block sequences
By James's Theorem, A. Ulger (Weak compactness in $L^{1}(\mu.X)$, Proc. Amer. Math. Soc. 113(1991),143-149.) proved that a bounded subset $A$ of a Banach space $X$ is relatively weakly compact if and ...
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gap in a Banach spaces ultrapower proof
This is an adaptation of a Heinrich proof, but I'm missing a key ingredient.
Conjecture. Suppose $(x_n)_{n=1}^\infty$ is a Schauder basis for a Banach space $X$ whose canonical isometric copy in $X^{*...
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Weak sequential compactness on the space of compact operators
Let $E,F$ be Banach spaces and let $A\subset K(E,F)$ be a subset of the space of compact operators from $E$ to $F$.
A result by Kalton states that $A$ is weakly compact if and only if $A$ is WOT* ...
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A weakly sequentially continuous operator which is not weakly continuous
I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.
So, let
$T$ an operator between a Banach space $X$ and itself.
$T$ is weakly ...
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Can quotient space be isomorphically isometric to some closed subspace of original one?
Suppose $\mathcal{B}$ is a Banach space which is not isomorphic to a Hilbert space. What condition should I pose on $\mathcal{B}$ such that for every closed subspace $\mathcal{M}$, the quotient space $...
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$C[0,1]$ fails the property (K)
Recall that a Banach space $X$ has the property (K) if every $w^{*}$-convergent sequence in $X^{*}$ admits a convex block subsequence which converges with respect to the Mackey topology. The property (...
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Measurable selection for argmin to distance
Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with ...
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Conditions on the inequality with a gauge norm
Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
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Weak topology on spaces of measures and Borel sets
Let $K$ be a compact Hausdorff space (not necessarily metric or even separable). Let $M(K)$ be the space of all Radon measures on $K$ (that is, finite signed regular Borel measures) endowed with the ...
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The $w^{*}$-convergent sequences and the Mackey topology on $X^{*}$
Let $X$ be a Banach space. Recall that the Mackey topology $\mu(X^{*},X)$ on $X^{*}$ is the topology of uniform convergence on weakly compact subsets of $X$. Let $(x^{*}_{n})_{n}$ be a sequence in $X^{...
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Sufficient condition for asymptotic-$\ell_{p}$ in terms of spreading models?
Let $(X,\|\cdot\|)$ be a Banach space with a Schauder basis and fix $p\in[1,\infty]$. Suppose that $X$ is asymptotic-$\ell_{p}$ with respect to this basis. It is known that the closed linear span of ...
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Weak relaxation of a strongly lower semi-continuous functional
Let $F$ be a lower semicontinuous functional on a Banach space $X$, wrt its strong topology. Is there a known form for the relaxation (lower semicontinuous envelope) of $F$ with respect to the weak ...
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When does an operator from $\ell_1$ to itself factor through $\ell_p$?
I would like to know whether a given operator from $\ell_1$ to itself, given by a matrix $A$, factors through $\ell_p$, for $p>1$.Does anyone know any references/results on this topic? I am ...
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History of the Lewis-Stegall theorem on factorization of representable operators
The following questions are about the history of a particular result in functional analysis, hence not "mathematical questions" per se; but I think they are relevant to the business of ...
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What are the complemented subspaces of $(\bigoplus\ell_q^n)_p$?
Bourgain/Casazza/Lindenstrauss/Tzafriri proved in their unconditional basis UTAP book (1985) that $\ell_1$ is the only nontrivial complemented subspace of $(\bigoplus\ell_2^n)_1$, and hence by duality ...
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Finding approximate eigenvectors: quantitative results
Let $X$ be a complex Banach space and $T \colon X \to X$ be a bounded operator. For every $x \in X \setminus \{0\}$, denote by $Y_x$ the smallest closed $T$-invariant subspace of $X$ containing $x$. ...
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A question on Grothendieck space
A Banach space $X$ is said to be Grothendieck if the weak and the weak* convergence of sequences in $X^{*}$ coincide. I have the following two questions.
Question 1. A Banach space $X$ is Grothendieck ...
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$C^j$-topology considered by Greene and Krantz
My question is about the $C^j$ topology used by Greene and Krantz in their paper "Deformations of Complex Structures, Estimates for the $\bar{\partial}$-equation, and stability of the Bergman ...
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Are there minimal topological conditions on a space 𝑋 for it to have a countable separating set?
Are there minimal topological conditions on a space $X$ for it to have a countable separating set?
A separating set here is a set $D \subset C(X)$ (where $C(X)$ is the space of continuous functions ...
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Why is density and separability needed for uniqueness of weak (time) derivatives?
Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if
$$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
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Equicontinuity-like property of a convex compact set
Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
Is there an ...
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Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
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Integral convergence with two sequences of functions
I came across this theorem just stated but has not proved and marked by 'it is easy to see'.
Theorem If $u_m$ and $v_m$ converges to $u$ and $v$ in $L^2([0,T];H^1(\Omega))$ weakly and $L^2([0,T];L^2(\...
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Ultrabornological representation for the space of uniformly continuous functions?
Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space
$$
C_{\omega_i}(\mathbb{R}^n,\...
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Banach embedding of finite dimensional spaces
Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]).
I am wondering whether are any result for the case when $r>s>2$?
...
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Connected components of bounded linear operators of $V = (\mathcal C(U(1), \mathbb C) , \lVert \cdot \rVert_\infty)$
This question is related to this one.
Consider the complex Banach space $V=(\mathcal C(U(1), \mathbb C), \Vert \cdot \Vert_\infty)$ where $\mathcal C(U(1), \mathbb C)$ is the space of continuous ...
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Algorithm/iterative procedure for constructing hypercyclic vectors?
Let $B$ be a separable Banach space and let $L:B\rightarrow B$ be a hypercyclic operator; here I use the definition of hypercyclicity given implicitly by Birkhoff's Transitivity Theorem: continuous ...
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Supremum over which sets makes $H^{\infty}$ non-separable?
It is known that the space $H^{\infty}$ of bounded holomorphic functions on the unit disk $D$ is non-separable with respect to the supremum norm $\|\cdot\|_{\infty}^{D}$. Let $E\subset D$ be connected ...
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With Khinchine's inequality, prove Fourier basis is unconditional in $L^{p}[0,1]$ only for $p=2$
I am trying to prove Problem 6.10 in "Classical and Multilinear Harmonic Analysis" by by Camil Muscalu and Wilhelm Schlag.
Problem
Problem 6.10. Let $1\le p < \infty$ and suppose that ...
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A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?
In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
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In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?
This is a cross-posted on MSE here.
Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
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Bounded weak and weak-$\star$ topologies and metrics
Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $\overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set
$$
d(x,y) := \sum_{n=1}^\infty 2^{-n} |(x-y, h_n)|...
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A subcontinuous function, which is not continuous
Let $E$ be a Banach space and $T: E\rightarrow E$ be a mapping. $T$ is said to be subcontinuous if for any sequences $(u_n)_{n\in\mathbb{N}}$ in $E$ that converge strongly to $u$ the sequence $(T(u_n)...
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On hereditarily reflexive Banach spaces
It was proved by W.B. Johnson and H.P. Rosenthal [Studia Math. 43 (1972), 77–92]
that every Banach space $X$ with $X^{**}$ separable is hereditarily reflexive:
every infinite dimensional closed ...
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Logical axioms used in the construction of counterexamples to ISP
In many cases, some problems are either solved in an affirmative way, or in a negative way. However, in some cases it turns out that some logical axioms lead to a proof of a certain statement, while ...
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Empty Weyl/Fredholm spectrum of an operator on an infinite dimensional Banach space
Let $X$ be a complex infinite dimensional Banach space, and let $T \in B(X)$ be nonscalar. The Fredholm spectrum of $T$ is defined by:
$$ \sigma_{\Phi} (T) := \lbrace \lambda \in \mathbb{C} : T- \...
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Concrete example of non-nuclear operator $E \to F$ and isometry $F \hookrightarrow G$ so that the composition $E \to F \hookrightarrow G$ is nuclear
DISCLAIMER: I posted the same question a week ago on Mathematics Stack Exchange.
We know by an abstract argument that there exist Banach spaces $E$, $F$, $G$ and maps $E \to F \hookrightarrow G$ such ...
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Exactness of injective tensor products
For (algebraic) tensor products, it is well-known that the functor $A\otimes_R \cdot:Mod_R\rightarrow Mod_R$ is only (left-) exact when $A$ is a flat $R$-module. In particular, all vector spaces are ...
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Renorming of $C[0,1]$ for a strictly convex dual
Let $C[0,1]$ be the space of all Real valued continuous functions on $[0,1]$ with the usual supremum norm. Does there exist an equivalent renorming on $C[0,1]$ such that the corresponding dual norm is ...
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Seminorm which is zero on dense subset
Let $X$ be a Banach space and let $\hat{X}$ be a dense subset of $X$. If $p$ is a seminorm on $X$ such that $p(x) =0 $ for all $x \in \hat{X}$, does $p(x) =0$ for all $x\in X$ (is $p$ the trivial ...
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Why the name `Lipschitz-Free Banach spaces'?
There are many names for the same objects that is known as the Arens--Eells spaces, transportation cost spaces, free Banach spaces over a (pointed) metric space, and Lipschitz-free Banach spaces.
The ...
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Gaussian integral $\int_X \|x\|_X^2 \mu(dx)$ in Banach space
For a centered Gaussian measure $\mu$ on a Hilbert space $X$, it is known that
$$\int_X \|x\|^2 \mu(dx) = tr(Q)$$ where $Q$ is the covariance operator. Is there a similar version for Gaussian measures ...
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Exterior algebra of normed spaces
This question is related to my prior question, but this one is aimed, even though it's more general. If $V$ is a vector space, we define the exterior algebra of $V$ do be:
$$\bigwedge V := \bigoplus_{...
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Any reflexive space has the property of Banach-Saks?
We say that a Banach space $(X,\|.\|)$ has the Banach-Saks property if every bounded sequence $(x_m)_m$ in $X$ admits a subsequence $(x_{m_n})_n$ which converges in the sense of Cesàro, that is, there ...
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Subspaces of $L_p([0,1])$ whose unit ball is compact for the topology of convergence in measure
Any information about the following questions would be welcome.
I wonder whether there are (well-known or easy) closed and infinite dimensional subspaces of $L_p([0,1])$ ($1<p<\infty$) whose ...
6
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The Calkin representation for Banach spaces
Let $X$ be an infinite dimensional Banach space. Let $\Lambda_{0}$ be the set of all finite dimensional subspaces of $X$ directed by the inclusion $\subseteq$. For each $\alpha\in \Lambda_{0}$, let $...