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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, $D \subset X^*$ ...
1 vote
1 answer
151 views

Is smoothness preserved under an isometric isomorphism?

Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
0 votes
0 answers
90 views

How to show a point is a weak* -weak continuous for the identity map on $X_1^*$ or on $X_1^{**}$?

I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak* -Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
2 votes
0 answers
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(...
3 votes
0 answers
156 views

Gowers' dichotomy for quotients

Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable. A ...
5 votes
1 answer
206 views

Compactness in trace class operators space

Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$. Are there easy ...
3 votes
1 answer
189 views

Ribe's Theorem: finitely representability between two uniformly homeomorphic Banach spaces

An infinite-dimensional Banach space $X$ is said to be crudely finitely representable (with constant $\lambda$) in an infinite-dimensional Banach space $Y$ if there is a constant $\lambda>1$ such ...
7 votes
2 answers
394 views

Tangent space to infinite dimensional manifolds

In finite dimensional geometry, there is a single invariant of a vector space - its dimension. This characterizes finite dimensional manifolds as being glued from Euclidean balls. This situation is ...
0 votes
0 answers
146 views

On the pointwise limit of a sequence of analytic functions

I have been confused with this problem for weeks now. Suppose I have Banach spaces $E$ and $F$ and a sequence of functions $f_{n}: U \subset E \to F$, where $U$ is open and nonempty. Let $x \in U$ be ...
2 votes
0 answers
82 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 $...
3 votes
2 answers
1k views

Extensions of Urysohn's inequality

A version of Urysohn's inequality states that for a symmetric convex body $K \subset \mathbb{R}^n$, one has $$ \left(\frac{\text{vol}(K)}{\text{vol}(B_2)} \right)^{1/n} \le \frac{1}{\sqrt{n}} E \; \| ...
4 votes
1 answer
132 views

Direct characterization of finite-dimensional $1$-injective Banach spaces

It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk ...
12 votes
3 answers
16k views

Dual space of $\ell^\infty$

Why can the elements of the dual space of $\ell^\infty(\mathbb N)$ be represented as sums of elements of $\ell^1(\mathbb N)$ and Null$(c_0)$? <hr: EDIT: As confirmed in the comments, the OP ...
9 votes
0 answers
163 views

Moore-Penrose partial isometries and hermitian elements

Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
7 votes
1 answer
415 views

Is there a “Closure-of-Range Theorem” for Banach spaces?

The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*$, the four conditions: $T(X)$ is $s$-closed; $T(X)$ is $...
1 vote
0 answers
87 views

Supremum of sums of functions in $L^1$ taking random signs

Consider the Banach space $X=L^1([0,1])$, and let $n\gg1$ and $x_1, ..., x_n$ be any points in the unit sphere of $X$. Is there any reasonable lower bound for $$\sup_{(\epsilon_i)_{i=1}^n \in \{-1,+1\}...
6 votes
1 answer
323 views

Hartogs' theorem in Banach spaces

In complex analysis one learns Hartogs' theorem: Let $U\subseteq \mathbb{C}^n$ open and $f: U \rightarrow \mathbb{C}$ a function. Then $f$ is analytic iff for all $1\leq i \leq n$ $$ z \mapsto f(...
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)^*...
0 votes
0 answers
50 views

About extreme case on complex interpolation

I'm trying to prove some equality of spaces via complex interpolation with the usual Calderon functor $[,]_\theta$. If $(E_0,E_1)$ is a compatible couple, it is known that $$[E_0,E_1]_j, j=0,1,$$ is a ...
10 votes
3 answers
739 views

Is there a version of Fischer-Riesz theorem for Banach space?

$( \Omega,F, P )$: a measurable space equipped with a finite measure $(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra $p$ : a constant bigger than $1$ ...
0 votes
0 answers
96 views

Sufficient condition for weak convergence in Banach spaces

The question is quite elementary but nonetheless no proof or counter example comes to mind immediately. Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ ...
7 votes
2 answers
248 views

Subspaces of $\ell_\infty^3$

Let $a,b\in\mathbb C$ be suc that $\max\{|a+b|,|a-b|\}\leq 1$ but $|a|+|b|>1.$ According to this paper by Arias, Figiel, Johnson and Schechtman https://www.jstor.org/stable/2155206?origin=crossref#...
4 votes
0 answers
148 views

Some questions on Hardy's spaces

In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
3 votes
2 answers
137 views

Non-complete space verifying uniform boundedness

Recently, I have seen the so-called uniform boundedness theorem, which says: Let $(X,∥⋅∥)$ be a Banach space and $(Y,∥⋅∥)$ be a normed linear space. Let $A⊂B(X,Y)$ be a pointwise bounded family of ...
2 votes
1 answer
136 views

Eigenvectors of the dual of positive irreducible operators

This question was previously posted on MSE. Let $E$ be a Banach lattice such that $E$ is an $M$-space. Assume that $T\colon E\to E$ is a positive bounded non-compact irreducible linear operator with ...
6 votes
1 answer
335 views

Existence of pairwise quasi-complementary but not complementary subspaces

Let $𝑋$ be an infinite-dimensional Banach space (complex or real). A subspace of $𝑋$ means a closed linear submanifold. Subspaces $M$ and $N$ of $X$ are quasi-complementary if $M\cap N=\{0\}$ and $M+...
0 votes
0 answers
78 views

What does analytic uniformly in $s$ mean?

Suppose I have a complex vector space $V$ with finite basis $\{e_{1},...,e_{s}\}$. Then, I can consider the algebra $\mathcal{U}$ of formal polynomials on the variables $e_{1},...,e_{s}$. Suppose ...
0 votes
1 answer
96 views

Existence of a complemented basic sequence

Let $X$ be an infinite-dimensional Banach space (complex or real). A subspace of $X$ means a closed linear submanifold. If $S$ is a non-empty subset of $X$, then $[S]$ denotes the closed linear span ...
0 votes
0 answers
49 views

Kadec-Klee property of an equivalent norm on a Hilbert space

Let us consider the space $\ell_2$ with the Hilbert norm $\Vert \cdot \Vert$ and consider the following eqivalent norm: $$ \Vert (r,x) \Vert_A^2 = \Vert (r, Tx)\Vert^2 + \max \{ \Vert x \Vert, \vert r ...
1 vote
2 answers
220 views

A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$

Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$. Note that $A$ is non-empty with a Baire category argument. I ...
3 votes
1 answer
376 views

A more general product rule for weak derivatives?

Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^...
2 votes
1 answer
244 views

Characterization of normed spaces based on violation of parallelogram law

For a normed linear space $(X, \|\cdot\|)$, the Jordan-von Neumann theorem specifies when exactly the norm is induced via an inner product, namely when the parallelogram law is satisfied. I would like ...
3 votes
0 answers
165 views

$S^{n}(V)$ is "approximately" $V$ when $n$ goes to infinity (in the setting of normed space)

Let $B$ be a (separable) Banach space. $(v_{i})_{i}, i\in\mathbb{N}$ being a family of linear independent vectors. $V$ being the span of $v_{i}$. I try to prove that $V$ is dense in $B$. I define a ...
11 votes
0 answers
342 views

The diagonal operators and unconditionality

The following is well-known: Theorem: Let $X$ be a Banach space with an unconditional basis $(e_n)_n$. Then the space of the diagonal operators with respect the basis $(e_n)_n$ endowed with the ...
4 votes
1 answer
158 views

Is the image of a complemented subspace complemented?

This question has been crossposted from mathstackexchange: Let $X, Y$ be two Banach spaces and $T:X\to Y$ a continuous surjection. Assume $Z$ is a complemented subspace of $X$ and that $T(Z)$ is ...
6 votes
1 answer
290 views

Analytic maps on Banach spaces: analyticity upgrade

Consider the following problem. Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and $$ f:U\to G $$ an analytic map, such ...
14 votes
2 answers
873 views

Which finite dimensional Banach spaces can be represented isometrically as spaces of bounded operators on a finite dimensional Hilbert space?

Background: It is known that every Banach space $X$ can be embedded isometrically as a subspace in the space $C(K)$ of continuous functions on a compact Hausdorff space $K$. Indeed, one can take $K$ ...
7 votes
2 answers
349 views

Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?

This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
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 ...
5 votes
1 answer
220 views

How big is the class of all closed range bounded linear operator?

Let $X$ and $Y$ be Banach spaces and let $CR(X,Y)$ denote the set $B(X,Y)$ of all bounded linear maps from $X$ to $Y$ with $T(X)$ closed in $Y$. Certainly $CR(X,Y)$ is not open in $B(X,Y)$ as given ...
11 votes
1 answer
227 views

Complemented subspaces of $C(\beta\mathbb N\times \beta\mathbb N)$

Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\...
3 votes
1 answer
207 views

Embedding of a Banach space in $\ell_\infty(\Gamma)$ with a subspace embedded in a copy of $\ell_\infty$

In this question (which I think may be interesting in its own) I was asking if we can find a copy of $\ell_\infty$ between a separable subspace $Y$ contained in $\ell_\infty(\Gamma)$ and the whole ...
3 votes
1 answer
169 views

Copy of $\ell_\infty$ inside $\ell_\infty(\Gamma)$ containing given subspace

To complete a proof I need to know if the following is true: Given a non-empty set $\Gamma$ and a separable subspace $Y$ of $\ell_\infty(\Gamma)$, there exists a subspace $A$ of $\ell_\infty(\Gamma)$ ...
5 votes
2 answers
432 views

Does closedness of the image of unit sphere imply the closed range of the operator

Let $X$ and $Y$ be Banach spaces and let $T:X\to Y$ be a bounded linear operator such that $T(S_X)$ is closed in $Y$. Does it imply that $T(X)$ is closed? Any hint is appreciated.
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, ...
1 vote
0 answers
105 views

Can a uniform weak null set of $c_0$ be uniformly embedded into a Hilbert space?

Remark that a uniform weak null set $A$ of $c_0$ satisfied that for any $\epsilon>0$ there exists a positive number $N(\epsilon)$ such that for every $f$ in $B(\ell_1)$, the unit ball of the ...
1 vote
1 answer
209 views

Rate of convergence of mollified functions in $L^p$ norm

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
0 votes
1 answer
106 views

Convergence of mollified functions in weighted $L^p$ norm

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
0 votes
1 answer
151 views

Super-reflexivity is separately determined

I've found this result that states that super-reflexivity is separably determined, i.e., if every separable subspace $Y\subset X$ of a Banach space is superreflexive then $X$ itself is superreflexive. ...
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 $\...

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