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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UMD constant of finite dimensional spaces
For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
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Converse to Riesz-Thorin Theorem
Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.
The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....
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votes
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answer
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Self-dual Orlicz sequence spaces
Suppose that $\ell_\phi$ is a reflexive Orlicz sequence space such that its dual space $\ell_\phi^*$ is isomorphic to $\ell_\phi$.
Is $\ell_\phi$ isomorphic to $\ell_2$?
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A separable Banach space and a non-separable Banach space having the same dual space?
I asked myself the following question when I was student just for curiosity. I asked a bit around (my professor, some researchers that I know), but nobody was able to give me an answer. So maybe it is ...
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2
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Connectedness of Invertible elements in a non- commutative C*- algebra
The Gelfand Naimark Segal theorem says that any complex C* algebra $A$ is isometrically isomorphic to a C* sub-algebra of bounded operators on a Hilbert space.
Here we see that the set of all ...
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Construction of Schauder bases on $C(X)$
Let $(X,d)$ be a compact metric space and let $C(X)$ be the set of continuous (bounded) real-valued functions on $X$ equipped with the usual supremum norm:
$$
\|f\|_{\infty}\triangleq \sup_{x\in X}|f(...
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answers
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Functions with smooth projections on finite-dimensional subspaces
Let $E,F$ be Banach spaces and $F$ be finite-dimensional and $E$ be strictly convex. Let $f\in C(F,E)$ have the property that:
$$
\text{For every finite-dimensional subspace $E'\subseteq E$ we have } ...
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Operator algebra on an invariant subset
In Rickart, page 50 Theorem 2.2.1, the statement is made: A linear subspace $\mathfrak{M}$ of the algebra $\mathfrak{A}-\mathfrak{L}$ is invariant with respect to the representation $a{\rightarrow}A_a^...
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Are stable images closed?
If $X$ is a Banach space and $T : X \to X$ is a continuous linear operator with the property that $T^{n}X$ equals $T^{n+1}X$ for some $n \ge 1$, does it follow that $T^{n}X$ is a closed subspace?
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Operator norm on tensor product of trace classes is multiplicative
Given Hilbert spaces $\mathcal H_1,\mathcal H_2,\mathcal K_1,\mathcal K_2$ and bounded linear maps $S_i:\mathcal B^1(\mathcal H_i)\to\mathcal B^1(\mathcal K_i)$, $i=1,2$ between the respective trace ...
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answer
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Closure of the space of Fredholm operators
Let $X,Y$ be two Banach spaces.
A bounded operator $A$ is Fredholm if $\ker A$ and $\mathrm{coker} A$ are finite dimensional. Denote by $Fred(X,Y) \subset \mathcal{L}(X,Y)$ the space of Fredholm ...
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Differentiability of the map $x\mapsto \delta_x$ in the Arens-Eells/Lipschitz-free space
$\DeclareMathOperator\AE{AE}\DeclareMathOperator\Lip{Lip}$Let $\AE(X)$ denote the Arens-Eells space on a Banach space $X$. Consider the map:
$$
\begin{aligned}
\delta: X & \rightarrow \AE(X)
\\
x&...
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answer
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$(1+\epsilon)$-injective Banach spaces, complex scalars
It is well known that a real Banach space which is $(1+\epsilon)$-injective for every $\epsilon >0$ is already 1-injective (Lindenstrauss Memoirs AMS, 1964, download here). Using common ...
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Standard function spaces with the approximation property
A Banach space $\mathcal{X}$ is said to have the approximation property (AP) if, for every compact set $K \subset \mathcal{X}$, there is a sequence of finite rank operators $\{U_n : \mathcal{X} \to \...
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How complex is the orbit equivalence relation of $\mathrm{Iso}_0(X)\curvearrowright S_X$ for $X=L^p([0,1])$?
For a Banach space $X$ let $S_X$ denote its unit sphere and let $\mathrm{Iso}_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\...
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Some estimates on tensor norms
Denote $M_n$ to be $n\times n$ matrix. For $X\in M_n$ define $\|X\|_1:=\max\limits_{1\leq j\leq n}\sum_{i=1}^n|x_{ij}|$ and $\|B\|_2:=\max\{|\sum_{i,j=1}^nb_{ij}x_iy_j|:|x_i|=|y_j|=1,\ 1\leq i,j\leq n\...
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Nonseparable counterexamples in analysis
When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...
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What is the smallest Lipschitz constant of a Lipschitz retraction of $\ell_\infty([0,1])$ onto $C[0,1]$?
By Theorem 1.6 in the book "Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss, the Banach space $C[0,1]$ is a Lipschitz retract of the Banach space $\ell_\infty[0,1]$. ...
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votes
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Operators "building" linear independant sets
Let $E$ be a separable Banach space and let $T\in L(E,E)$.
Is there a condition on $T$ ensuring that:
$$
\mbox{$\{x_n\}_{n=1}^N\subseteq E$ is linearly independent} \Rightarrow
\{T(x_n)\}_{n=1}^N\cup \...
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1
answer
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Banach space containing uniformly complementend $\ell_p^n$s
Let $X$ be a Banach space such that both $X$ and $X^*$ have finite cotype. Also assume that $X$ is an inductive limit of finite dimensional Banach spaces $X_n\subseteq X_{n+1}.$ Fix $1<p<\infty.$...
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Banach space with dual not a GT space
Let $X$ be a Banach space. A bounded linear map $u:X\to\ell_2$ is said to be $1$-summing if for all finite sequence $(x_i)\subseteq X$ there is a constant $C>0$ such that $\sum\|ux_i\|\leq C\sup\...
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Duality $(M/N)^*\equiv N^\perp/M^\perp$ for closed subspaces $N\subset M$ of a Banach space
Let $M$ be a closed subspace of a Banach space $X$. Then we can identify $(X/M)^*$ with $M^\perp$ and $M^*$ with $X^*/M^\perp$.
Indeed, if $Q^*:X\to X/M$ is the quotient map, then $Q^*:M^*\to X^*$ is ...
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Need reference of books/papers which deals with Ternary Banach Algebras
I'm interested in learning about ternary Banach Algebras ( mainly ideal theory and tensor product)
Can someone please recommend me some papers/ books/ notes which deals with mentioned topics?
Thank ...
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Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?
I know that $\ell_2$ is isomorphic to a subspace of $L_p(0,1)$ for any $1\le p<\infty$. However, I haven't seen anything about $L_\infty$. Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?...
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Subprojective Orlicz sequence spaces
A Banach space $X$ is subprojective if every infinite dimensional closed subspace $Y$ of $X$ contains an infinite dimensional subspace $Z$ which is complemented in $X$.
I am interested in conditions ...
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When does a positive operator preserve invertibility
Let $\Omega_1,\Omega_2$ be compact Hausdorff spaces and let $P:C(\Omega_1)\longrightarrow C(\Omega_2)$ be a unital positive operator. I wanted to know if there is a necessary and sufficient condition ...
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Biorthogonal weakly null basic sequences
Let $E$ be a Banach space, let $e_{n}\in E$ and $g_{n}\in E^{*}$ be biorthogonal basic sequences (i.e. $\left<e_n,g_m\right>=\delta_{mn}$ ). Moreover, both of these sequences are weakly null. (...
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Semi-norms on LCS inductive limit of Banach Spaces
Let $(E_n,i_n)_{n\in\mathbb{N}}$ be an direct system of Banach spaces in the category of locally convex spaces (LCSs) with continuous linear maps and let $E_{\infty}$ by their inductive limit. What ...
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1
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Reference request: Baire's theorem for operator ranges
Let $F$ be a Banach space. A vector subspace $U \subseteq F$ is called an operator range if there exists a Banach space $E$ and a bounded linear mapping $T: E \to F$ such that $TE=U$. By a quotient ...
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Hamel basis with all coordinate functionals discontinuous
If $X$ is an infinite dimensional (separable) Banach space, can one find a Hamel basis $(x_\alpha)_{\alpha\in\Lambda}$ such that all coordinate functionals $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ ...
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Regarding an element being self adjoint
Let $A$ be a unital C*-algebra. Let $x,y\in A$ be self adjoint elements in $A$, with $x$ being invertible. Can we say that the spectrum of $x^{-1}y$ is a subset of the real line? I know this is true ...
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Convergence of Gaussian measures
Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian ...
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Space of compact operators
I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach ...
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votes
1
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Uniform boundedness principle for almost surely converging sequence of operators
I'd like to do the following: I consider a separable Banach space $X$ with a probability measure $\mu$ on the Borel $\sigma$-algebra $\mathcal B(X)$. Additionally, I have a sequence of measurable, ...
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3
votes
1
answer
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G.L. l. u. st. for subspaces of Banach spaces with an unconditional basis
A Banach space $X$ has Gordon-Lewis local unconditional structure (G.L. l. u. st.) if for every finite dimensional subspace $E$ of $X$, the inclusion operator $i:E\to X$ factors through a finite ...
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$K$-convex Banach spaces
Let $X$ be a Banach space. We say that $X$ contains $\ell_1^n$'s uniformly iff for all $n\in\mathbb N$ there exist subspaces $X_n\subseteq X$ with $d(X_n,\ell_1^n)\leq \lambda$ for some $\lambda\geq 1$...
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For which Banach spaces is the self composition operator Lipschitz?
Let $X\subseteq \{f|f:D\rightarrow \mathbb{R}^n\}$ be a Banach space, with at least all polynomials on $D$ contained in $X$, where $D\subseteq \mathbb{R}^n$ is open and bounded.
Let $U\subseteq X\cap \...
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Another question about asymptotic models in Banach spaces
The array $(x_{i}^{k})_{i=1,k\in\mathbb{N}}^{\infty}$ of normalized $M$-basic sequences in a Banach space $X$ is itself called $M$-basic if, for every $k\leq i_{1}<i_{2}<\ldots$, the diagonal ...
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Density of signed measures in dual space
Let $B$ be a Banach space of functions on a Radon space $X$. By the Hahn-Banach theorem, we know that the canonical evaluation map is isometric. That is, for every $f \in B$, we have
$$\|f\| = \sup_{\...
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Dual of isometric copies into dual Banach spaces
Let $X$ be a Banach space and $X_1\xrightarrow{}X$ isometrically. Under some assumption can we guarantee that $X^*$ contains an isometric copy of $X_1^*$. I am also interested to know if this happens ...
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Certain decompositions of decomposable Banach spaces
Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another ...
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Property $(V_1)$ for Banach spaces
This aim of this note is to record a problem that still seems to be open.
Räbiger, in his doctoral thesis, defined property $(V_1)$ as follows: A Banach space $X$ has property $(V_1)$ if every ...
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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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answer
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Complemented subspaces of $\mathcal{L}_{p}$-spaces
In 1968, J. Lindenstrauss and A. Pe{\l}czy'{n}ski posed a problem: Is every complemented subspace $X$ of an $\mathcal{L}_{p}$-space ($1\leq p\leq \infty$) either an $\mathcal{L}_{p}$-space or ...
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vote
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answer
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Every complemented subspace of a $C(T)$-space is an $\mathcal{L}_{\infty}$-space
Let $T$ be a compact Hausdorff space and $X$ be an infinite-dimensional complemented subspace of $C(T)$.
Question 1. Assume that $X$ has a subspace $U$ that is isomorphic to $c_{0}$. Given any ...
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4
votes
1
answer
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On the intersection of two Orlicz spaces
It is well-known that if $1\leq p\leq q\leq \infty $ then
$$ L^p(X)\cap L^q(X)\subset L^r(X)\quad\quad \text{whenever $r\in [p,q]$}\tag{I}\label{Eq}.$$
Indeed let $u\in L^p(X)\cap L^q(X)$. For some $...
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1
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Closedness of the image of the unit ball
Let $X$ be a Banach space and let $P$ be a bounded, linear projection on $X$. Is $P[B_X]$ closed in $X$? Here $B_X$ is the closed unit ball of $X$.
This is trivial if $X$ is reflexive, but otherwise ...
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1
answer
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When is a function a convex conjugate?
Let $X$ be a Banach space; $X^*$ be its dual; and $g:X^*\to\mathbb R\cup\{\infty\}$ be a proper, convex, weak${}^*$-lower semicontinuous function with weak${}^*$-compact effective domain.
Question: Is ...
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votes
1
answer
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Example of empty projection in strictly convex Banach space
Let $X$ be a strictly convex Banach space, let $C\subseteq X$ be a nonempty closed convex set, and let $P_C$ be the set-valued metric projection
$$P_C(x) = \{y\in C : \|x-y\| = d(x,C)\} . $$
We know ...
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votes
1
answer
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Finite-dimensional subspaces of $c_{0}$
Let $M$ be a finite-dimensional subspace of $c_{0}$, and let $\varepsilon>0$.
Question. Does there exist a finite rank projection from $c_{0}$, of norm $\leq 1+\varepsilon$, onto a subspace $N$ of ...
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