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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
1 vote
1 answer
153 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 ...
Tuh's user avatar
  • 113
5 votes
1 answer
208 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 ...
lulli_'s user avatar
  • 59
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:...
Tuh's user avatar
  • 113
3 votes
0 answers
159 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 ...
M.González's user avatar
  • 4,461
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 ...
Xiangbo's user avatar
  • 33
7 votes
2 answers
397 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 ...
0x11111's user avatar
  • 593
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 ...
InMathweTrust'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
  • 4,461
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 $...
Pietro Majer's user avatar
  • 60.6k
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 ...
Hannes Thiel's user avatar
  • 3,497
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 ...
Martin Argerami's user avatar
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\}...
HHN's user avatar
  • 393
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+...
Janko Bracic's user avatar
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 ...
Tomas smith Smith's user avatar
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<...
A beginner mathmatician's user avatar
2 votes
1 answer
246 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 ...
Hikaru's user avatar
  • 213
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 $(x_n,y)$ ...
an_ordinary_mathematician's user avatar
1 vote
2 answers
221 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 ...
Ali Taghavi's user avatar
3 votes
1 answer
377 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^...
Bogdan's user avatar
  • 1,759
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.
Anupam's user avatar
  • 585
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
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 ...
Janko Bracic's user avatar
7 votes
2 answers
351 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 ...
David Gao's user avatar
  • 2,830
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 ...
S Argyros's user avatar
  • 986
5 votes
1 answer
221 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 ...
Anupam's user avatar
  • 585
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 ...
Pelota's user avatar
  • 655
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 ...
Ken.Wong's user avatar
  • 523
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 ...
InMathweTrust's user avatar
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 ...
Esteban Martínez's user avatar
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 ...
Akira's user avatar
  • 825
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 $\ell_\infty(\Gamma)$ ...
Esteban Martínez's user avatar
6 votes
1 answer
291 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 ...
Lorenzo Pompili's user avatar
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
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. ...
Michelangelo's user avatar
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
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 ...
Akira's user avatar
  • 825
2 votes
1 answer
148 views

Is projection of a closed subspace Borel?

Specifically, letting $E$ be a separable infinite-dimensional real Banach space, and $D_2$ in $E\times E$ a closed linear subspace, is then $\{\,x:\exists\,y\,;(x,y)\in D_2\}$ a Borel set in $E\,$? ...
TaQ's user avatar
  • 3,584
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 ...
yu tianfeng's user avatar
7 votes
2 answers
345 views

Integral means vs infinite convex combinations

Let $(X,\mathcal A, \mu)$ be a probability space, $\mathbb E$ a Banach space, and $f:X\to\mathbb E$ a Bochner integrable function. Does there exist a sequence $(x_k)_{k\ge 1} $ in $X$, and a ...
Pietro Majer's user avatar
  • 60.6k
2 votes
1 answer
136 views

Is there a scalar product which makes orthonormal the family of complex functions $ (f_n)_{ n \geq 1 } $?

Let $ (f_n)_{ n \geq 1 } $ be a family of complex functions defined as follow, $ \forall n \geq 1 $, $$ f_n (z) = \dfrac{1}{n^{z}} $$ I would like to ask you if it is possible to construct a ( non-...
Angel65's user avatar
  • 595
7 votes
3 answers
909 views

Using the Stone-Weierstrass theorem to solve an integral limit

The following question was posted on math stack exchange here but it got no answers Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
Shthephathord23's user avatar
2 votes
0 answers
185 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

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
5 votes
1 answer
562 views

interiors of positive cones in ordered Banach spaces

I have a couple of questions about ordered Banach spaces and interiors of their positive cones. I would appreciate your insights and any recommended references. I want to know several examples of ...
Saito's user avatar
  • 79
0 votes
1 answer
318 views

A variation of the Riesz Lemma

Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
Emerick's user avatar
  • 153
-1 votes
1 answer
139 views

$L^1$ convergence

Setting For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
Anthony's user avatar
  • 125
0 votes
1 answer
163 views

Counterexample wanted: Banach space but not BK-space

What is an example of a Banach space that is not a BK-space? A normed sequence space $X$ (with projections $p_n$) is a BK Space if $X$ is Banach space and for all natural numbers $n$, $p_n(\bar{x}) = ...
Maulana's user avatar
4 votes
1 answer
165 views

Dual spaces of Banach-valued $L^{p}$-spaces

Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\...
G. Blaickner's user avatar
  • 1,429
5 votes
1 answer
188 views

On a property for normed spaces

I asked this question on Math Stackexchange, but I didn't get an answer: https://math.stackexchange.com/questions/4881155/on-a-property-for-normed-spaces?noredirect=1#comment10410489_4881155 I came ...
Markus's user avatar
  • 1,361

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