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Embedding a normed space as a hyperplane

Let $X$ be a real normed space and suppose that $X$ is a closed hyperplane of a bigger space $\tilde X$. Given any unit vector $u$ in $\tilde X\setminus X$, consider the function $p:X\to\mathbb R$ ...
Black's user avatar
  • 483
1 vote
0 answers
45 views

Generalizations of the Wiener Tauberian Theorem to Musielak-Orlicz spaces

Musielak-Orlicz spaces provide a generalization of the usual $L^p$ spaces on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to spaces of functions for which the Luxemburg norm $$ \|f\|_M:=\inf\left\{\lambda &...
ABIM's user avatar
  • 5,405
0 votes
0 answers
106 views

A noncontinous algebra map between Banach algebras

What is an example of two Banach algebras $A$ and $B$, and an algebra map $\phi:A \to B$ which is not continuous?
Dick Johnson's user avatar
3 votes
0 answers
145 views

Non uniqueness of center of the Banach-Mazur compactum

In "The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization" Szarek and Bourgain prove a proportional Dvoretzky-Rogers factorization : Given $1>\delta>0$ , there ...
Gericault's user avatar
  • 245
2 votes
1 answer
183 views

Does set of finitely additive probability measures embed linearly into a strictly convex dual Banach space?

I am trying to better understand a condition that appears in Theorem 1 of this paper. Let $K$ be a convex and compact subset of a locally convex tvs. The condition is: $K$ embeds linearly into a ...
aduh's user avatar
  • 869
1 vote
1 answer
164 views

Asymptotic models and passing to sub-arrays

If, for all $k\in\mathbb{N}$, $(x_{i}^{k})_{i=1}^{\infty}\in X^{\mathbb{N}}$ is normalized and $M$-basic and if, in addition, for all $k\leq i_{1}<i_{2}<\ldots$ the diagonal sequence $(x_{i_{k} }...
JWP_HTX's user avatar
  • 201
9 votes
1 answer
333 views

Closedness of linear image of positive L1 functions

Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\...
e.lipnowski's user avatar
4 votes
2 answers
231 views

Inclusion of infinite intersection

Let $E$ be a Banach space, $T:E\rightarrow E$ a continuous bounded nonlinear mapping., and $\{x_n\}_{n\in\mathbb N}$ such that $$x_{n+1}=T(x_n),\:\forall n\in \mathbb{N}:=\{0,1,\cdots\}.$$ Let $$X_n=\...
Motaka's user avatar
  • 291
3 votes
1 answer
220 views

An improvement of Sobczyk's Theorem

Sobczyk's theorem states that if a separable Banach space $X$ contains a subspace isometric to $c_{0}$, then $X$ contains a subspace $Z$ which is isometric to $c_{0}$ and is $2$-complemented in $X$. ...
Dongyang Chen's user avatar
9 votes
1 answer
608 views

Interpolation theory and $C^k$-spaces

Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
Jan Bohr's user avatar
  • 779
4 votes
0 answers
179 views

Condition on kernel convolution operator

I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with $$ \Phi_i(2t)\approx \Phi_i(t), \quad i=1,2 $$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
Forbs's user avatar
  • 101
6 votes
2 answers
201 views

holomorphy in infinite dimensions (holomorphic families of operators)

Let $X$ be a Banach space (over $\mathbb C$), and let $\mathcal L(X)$ be its algebra of bounded linear operators. Let $U\subset \mathbb C^N$ be an open subset, and $f:U\to \mathcal L(X)$ a function ...
André Henriques's user avatar
9 votes
1 answer
355 views

Scottish Book Problem 172

The problem is formulated using old terminology and I want to understand what it actually says. The problem reads: "A space $E$ of type (B) has the property (a) if the weak closure of an ...
Hannes Thiel's user avatar
  • 3,497
4 votes
1 answer
604 views

Weak convergence in a product space

Given a function $f: Y\longrightarrow Y$ ($Y$ is a Banach space). Assume that $f$ satisfies: If $y_n \rightharpoonup y $, then $f(y_n)\rightharpoonup f(y) \text{ in } Y$; $f$ is weakly compact; ...
Malik Amine's user avatar
3 votes
1 answer
148 views

Measurability of superposition operator with non-separable Banach space

Let $f\colon I \times X \to \mathbb{R}$ be a map where $I \subset \mathbb{R}$ is an interval, $X$ is a Banach space (possibly non-separable) and we have $$t \mapsto f(t,x) \text{ is measurable}$$ $$x \...
MMML's user avatar
  • 107
5 votes
1 answer
1k views

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 ...
Black's user avatar
  • 483
1 vote
1 answer
92 views

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}...
JWP_HTX's user avatar
  • 201
3 votes
1 answer
170 views

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 ...
Curious's user avatar
  • 143
1 vote
1 answer
184 views

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-...
user124297's user avatar
2 votes
1 answer
215 views

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^{*...
Ben W's user avatar
  • 1,591
4 votes
1 answer
281 views

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* ...
waldrop's user avatar
  • 103
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
1 vote
0 answers
73 views

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^{...
Dongyang Chen's user avatar
3 votes
1 answer
176 views

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 ...
JWP_HTX's user avatar
  • 201
4 votes
0 answers
146 views

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 ...
Gamabunto's user avatar
8 votes
0 answers
196 views

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 ...
Yemon Choi's user avatar
  • 25.8k
4 votes
0 answers
75 views

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 ...
Ben W's user avatar
  • 1,591
6 votes
2 answers
509 views

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 ...
Dongyang Chen's user avatar
5 votes
1 answer
159 views

$C^j$-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 ...
Pita's user avatar
  • 113
5 votes
1 answer
153 views

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^\...
StopUsingFacebook's user avatar
2 votes
1 answer
70 views

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 ...
erz's user avatar
  • 5,529
0 votes
0 answers
45 views

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 ...
ABIM's user avatar
  • 5,405
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
0 votes
1 answer
81 views

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,\...
ABIM's user avatar
  • 5,405
3 votes
1 answer
138 views

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 $r>s>2$? ...
user92646's user avatar
  • 617
1 vote
0 answers
115 views

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 ...
ABIM's user avatar
  • 5,405
4 votes
1 answer
293 views

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 ...
erz's user avatar
  • 5,529
15 votes
2 answers
2k views

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-...
Nikhil Sahoo's user avatar
  • 1,225
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
9 votes
1 answer
242 views

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 ...
M.González's user avatar
  • 4,461
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
1 vote
0 answers
74 views

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- \...
Manuel Norman's user avatar
4 votes
1 answer
171 views

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 ...
J. van Dobben de Bruyn's user avatar
3 votes
1 answer
406 views

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 ...
ABIM's user avatar
  • 5,405
4 votes
1 answer
406 views

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 ...
Tanmoy Paul's user avatar
3 votes
1 answer
339 views

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 ...
Shash's user avatar
  • 41
1 vote
0 answers
277 views

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 ...
A_curious_asker's user avatar
1 vote
0 answers
139 views

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 ...
Made's user avatar
  • 115
8 votes
1 answer
207 views

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 ...
user159631's user avatar

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