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Classical subspaces of non-atomic Banach lattices

Tsirelson's space was the first example of a Banach space which does not have a subspace isomoprhic to any of the classical spaces $\ell_p$, $1\leqslant p<\infty$, or $c_0$. As this space has a $...
user avatar
3 votes
0 answers
422 views

Isometries between subspaces of finite-dimensional vector spaces

I would like to characterise the subspaces of $\ell_p^n(\mathbb{R})$ that are isometric (for $p$ an even integer). In the literature, I have found few results related to this. Taking $n \le m$, one ...
Dave's user avatar
  • 31
3 votes
0 answers
89 views

Discrete Lions Peetre interpolation

In S. Heinrich's "Closed operator ideals and interpolation," Heinrich describes two equivalent descriptions of Lions-Peetre interpolation space $(X,Y)_{\theta, p}$ for $0<\theta<1$ and $1\...
user avatar
3 votes
0 answers
125 views

Commutative discrete cyclic operator groups on topological vector spaces

Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
Bedovlat's user avatar
  • 1,959
3 votes
0 answers
169 views

A spanning set for an annihilator set on a Banach space

Let $(z_n)$ be a $H^\infty$-interpolating sequence on the open complex unit disc $\mathbb D$. If $A$ is some Banach space of analytic functions on the disc, denote by $X$ the closed subspace of all ...
user106480's user avatar
3 votes
0 answers
243 views

A universal operator between separable Banach spaces

The Banach space $C[0,1]$ is universal for all separable Banach spaces in the sense that for a separable Banach space $X$ there is an isometric isomorphism from $X$ into $C[0,1]$. My question is ...
Kevin Beanland's user avatar
3 votes
0 answers
104 views

independent symmetric 3-valued random variables in Lp

Consider the following excerpt from this paper: Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...
Ben W's user avatar
  • 1,591
3 votes
0 answers
168 views

Norm condition in a Banach lattice

Consider the following "condition (J)" on the norm of a (real or complex) Banach lattice $E$: whenever $x$ and $y$ are disjoint (i.e., $|x|\wedge |y|=0$) then $\|x+y\|+\|x-y\|=2\|x\|+2\|y\|$. ...
Fred Dashiell's user avatar
3 votes
0 answers
642 views

question about bidual normed space

Consider a Banach $\mathbb{R}$-space E and an element $u\in E''$ such that : for all sequence $\phi_n\in E'$ which $\sigma(E',E)$-converges to $\phi \in E'$, one has $\lim u(\phi_n)=u(\phi)$. Is it ...
Terry Grashin's user avatar
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0 answers
163 views

Isometric automorphism of $c_0$ different than coordinate permutation

Does there exist an isometric automorphism of $c_0$ which is not a permutation of coordinates?
robibok's user avatar
  • 311
2 votes
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63 views

Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$

In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
Józef Zápařka's user avatar
2 votes
0 answers
83 views

The support of the functions in the closed span of the Rademacher functions in $L_1(0,1)$

Given a measurable function $f:(0,1)\to \mathbb{R}$, we denote by $M(f)$ the measure of the set $\{t\in (0,1) : f(t)\neq 0\}$. It is not difficult to prove that if $(f_n)$ is a normalized sequence in $...
M.González's user avatar
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2 votes
0 answers
47 views

Norm density of evaluation functionals in the space of weak$^*$ continuous multilinear functionals on products of dual Banach spaces

Let $K$ be a compact (metrizable) space and let $C(K)$ be the Banach space of continuous real-valued functions on $K$, equipped with the supremum norm. It is then well known that the dual space $C(K)^*...
kiliroy's user avatar
  • 56
2 votes
0 answers
82 views

What is Lipschitz constant of the radial renormalization $(X,\|\cdot\|_a) \rightarrow (X,\|\cdot\|_b)$ on a normed vector space $X$

Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping $$ f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X, $$ with $f(0)=0$ is a radial and maps ...
shuhalo's user avatar
  • 5,327
2 votes
0 answers
96 views

Isometric Schröder-Bernstein theorem for injective Banach spaces?

It's known that every injective Banach space is of the form $C(M)$ where $M$ is a compact, Hausdorff, extremally disconnected topological space. Let $X$, $Y$ be two injective Banach spaces such that, ...
Onur Oktay's user avatar
  • 2,605
2 votes
0 answers
88 views

Dependence and $L^2$ projections of functions

tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function? Let $w$ be a density on $\...
shawn532's user avatar
2 votes
0 answers
184 views

Example of space which is weak Hahn-Banach smooth but not Hahn-Banach smooth

A Banach space $X$ is said to be Hahn-Banach smooth if every linear functional on $X$ has a unique norm-preserving extension over $X^{**}$. Weak Hahn-Banach smoothness is what if the above condition ...
Tanmoy Paul's user avatar
2 votes
0 answers
88 views

An example of an $\mathcal{L}_\infty$ Banach space with property p-(V) and without property (V)

Here are the definitions for property $p$-$(V)$ and property $(V)$. A Banach space $X$ has property $(V)$ if and only if every unconditionally converging operator $T$ from $X$ to any Banach space $Y$ ...
Ioana Ghenciu's user avatar
2 votes
0 answers
78 views

Array-determined operator ideals

For a Banach space $X$, we, of course, know what it means for a sequence to be weakly null (to converge to zero in the weak topology). An array in the Banach space $X$ is a sequence of sequences, $(...
jwhite's user avatar
  • 121
2 votes
0 answers
98 views

Geometric interpretation of uniform convexity condition

I first want to recall the moduli of uniform smoothness (US), uniform convexity (UC), asymptotic uniform smoothness (AUS), and asymptotic uniform convexity (AUC). Throughout, let $X$ be an infinite ...
user516424's user avatar
2 votes
0 answers
78 views

Analogy between quasi-injective modules & extensible Banach spaces

Let $X$ be a module. $X$ is said to be quasi-injective if every homomorphism $h:A\to X$ from any submodule $A\subseteq X$ has an extension to an endomorphism $\tilde{h}:X\to X$. A module $X$ is quasi-...
Onur Oktay's user avatar
  • 2,605
2 votes
0 answers
79 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 ...
Akira's user avatar
  • 825
2 votes
0 answers
148 views

Quotients of $c_0\mathbin{\hat{\otimes}_{\pi}}\ell^1$

Let $\hat{\otimes}_{\pi}$ denote the projective tensor product. Let $$\mathcal{S} = \{V\subseteq c_0\mathbin{\hat{\otimes}_{\pi}}\ell^1\textrm{ closed subspace}: {c_0\mathbin{\hat{\otimes}_{\pi}}\ell^...
Onur Oktay's user avatar
  • 2,605
2 votes
0 answers
354 views

Weakly null sequences in projective tensor products

First, I'd like to record a question that may still be open. The snippet below is taken from DiestelPuglisi2009. Second, let $E$ be a Banach space, $(u_n)$ be a weakly null sequence in the projective ...
Onur Oktay's user avatar
  • 2,605
2 votes
0 answers
55 views

The initial sigma-algebra on the dual of a Banach lattice

Let $E$ be an AL space (i.e. a Banach lattice whose norm is additive on the positive cone $E_+$) that satisfies Mazur's condition (every sequentially weak$^*$-continuous functional on $E'$ is weak$^*$-...
HardyHulley's user avatar
2 votes
0 answers
76 views

Fractional integration in Orlicz spaces

I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley. And I would like to understand one question: Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...
user124297's user avatar
2 votes
0 answers
140 views

Extracting block sequences from $\ell_p^n$-like sequences in isomorphic copies of $\ell_p$

One of the characterizing properties of the canonical bases of the $\ell_p/c_0$ spaces is their perfect homogeneity (Zippin), so that every normalized block sequence in these spaces behaves like the ...
user469053's user avatar
2 votes
0 answers
39 views

Example when Lorentz-Shimogaki condition satisfied with a specific Young's function

Let $\Psi(t)=\int_0^t\psi(s)$ be a Young's function. Then, $\Psi$ satisfies the Lorentz-Shimogaki condition if $$ \int_0^{\infty}\frac{\Psi(st)}{v(t)^2}\psi(t)dt< \infty. $$ Denote $\rho_{\Psi}=\...
user124297's user avatar
2 votes
0 answers
134 views

Fourier type of asymptotic-$\ell_{2}$ Banach spaces

A Banach space $X$ is said to have Fourier type $p\in[1,2]$ if the Fourier transform $\hat{f}(s):=\int_{\mathbb{R}}e^{-ist}f(t)dt$ defines a bounded linear operator from $L_{p}(\mathbb{R},X)$ to $L_{p'...
JWP_HTX's user avatar
  • 201
2 votes
0 answers
168 views

On weak Hahn-Banach smoothness

Let us recall Phelp's property-$U$: A subspace $Y\subset X$ is said to have property-$U$ if every $y^*\in Y^*$ has unique norm preserving extension over $X$. $Y$ is weak Hahn-Banach smooth if $y^*$ ...
Tanmoy Paul's user avatar
2 votes
0 answers
97 views

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?
Andy Hammerlindl's user avatar
2 votes
0 answers
85 views

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 } ...
ABIM's user avatar
  • 5,405
2 votes
0 answers
89 views

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 ...
JWP_HTX's user avatar
  • 201
2 votes
1 answer
431 views

Density of $w^*$-support points

I am looking for a simple proof of the following theorem — wasn't able to come up with one myself. Should be a use of the Bishop–Phelps theorem, in some way: Let $X$ be a Banach space, $D \subset X^*$ ...
Tomer's user avatar
  • 165
2 votes
0 answers
57 views

Is this Beppo-Levi curl space a Banach space?

Let us define the quotient space: $$ V = \{ \mathbf{u} \in L^2_{loc}(\mathbb{R}^3; \mathbb{R}^3) : \operatorname{curl} \mathbf u \in L^2(\mathbb{R}^3; \mathbb{R}^3) \} / \nabla H^1_{loc}(\mathbb{R}^3)....
GaC's user avatar
  • 163
2 votes
0 answers
62 views

Decomposition of the Orlicz norm into sequential norm

I am bearing seeking for a sequential decomposition of the norm in Orlicz space. Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$. Given $u\in L^p(\Bbb R^d)$ let $$n\...
Guy Fsone's user avatar
  • 1,101
2 votes
0 answers
347 views

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 $...
JohnLee's user avatar
  • 53
2 votes
0 answers
137 views

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_{\...
user124297's user avatar
2 votes
0 answers
83 views

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(\...
Lev Bahn's user avatar
  • 239
2 votes
0 answers
1k views

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)|...
Jorge E. Cardona's user avatar
2 votes
0 answers
129 views

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 ...
Manuel Norman's user avatar
2 votes
0 answers
159 views

Explicit homeomorphism between $L^p$ and Sobolev Space

From the Anderson-Kadec theorem, we know that all separable infinite-dimensional Banach spaces are homeomorphic. I'm wondering, is there an explicit such homeomorphism between $W^{p,k}(\mathbb{R}^n)$ ...
ABIM's user avatar
  • 5,405
2 votes
0 answers
520 views

Example of a non-reflexive Banach space and two sequences

Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$. If $X$ is reflexive, ...
Karim KHAN's user avatar
2 votes
0 answers
89 views

A quantitative characterization of bounded approximation property

Recall that a Banach space $X$ has the approximation property (AP for short ) if for every compact subset $K$ of $X$ and every $\varepsilon > 0$, there exists a finite rank operator $S$ on $X$ such ...
Dongyang Chen's user avatar
2 votes
0 answers
69 views

Banach lattices $X$ for which $L_p(\mu)\subset X$ or $X\subset L_q(\mu)$

It is well known (see vol. II of Lindenstrauss and Tzafriri's book) that an order continuous Banach lattice $X$ with a weak unit admits a representation as a (in general not closed) ideal of $L_1(\mu)$...
M.González's user avatar
  • 4,461
2 votes
0 answers
109 views

Quotient Banach space whose dual map sends the ball onto a given convex subset

Let $X$ be a Banach space and let $A$ be a closed, convex and balanced subset of $B_{X^{*}}$ (where $B_{X^{*}}$ denotes the closed unit ball of the dual $X^{*}$). Is there a closed subspace $M$ of $X$ ...
Dongyang Chen's user avatar
2 votes
0 answers
55 views

A holomorphic map into a Hilbert space with prescribed orthogonality

This is a variation of my previous question. Let $X\subset \mathbb{C}^n$ be a domain, and let $L:X\times X\to \mathbb{C}$ be such that $L(x,x)>0$, $L(y,x)=\overline{L(x,y)}$ and $L(\cdot,y)$ is ...
erz's user avatar
  • 5,529
2 votes
0 answers
326 views

Dual of the space of affine functions

Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\...
Tanmoy Paul's user avatar
2 votes
0 answers
149 views

Projection semigroup of an isolated eigenvalue

I'm currently working with a paper and I don't get something there. Let $A$ be a closed operator on a Banach space $X$ and $\lambda \in \sigma(A)$ an isolated eigenvalue, i.e. there is a $r > 0$ ...
Yaddle's user avatar
  • 381
2 votes
0 answers
124 views

Logarithm of $L^p$ space

I encountered the following space as a natural space for setting up a certain problem: $$ S_m^p = \{f \colon I \to \mathbb{R} \text{ measurable }; m^{f} \in L^p(I)\} $$ Here, $I$ is an open bounded ...
Tommi's user avatar
  • 648

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