**0**

votes

**1**answer

330 views

### Norms agreeing on dense subspace [closed]

Suppose $(B,\|\cdot\|)$ is a Banach space, $V\subset B$ a dense subspace, and $V$ is equipped with a norm $\|\cdot\|_V$ such that $\|x\|_V = \|x\|$ for all $x\in V$.
Is $(B,\|\cdot\|)$ a completion ...

**3**

votes

**0**answers

402 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 ...

**0**

votes

**1**answer

195 views

### Absolute norms and 1-unconditional sums

Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, ...

**4**

votes

**1**answer

282 views

### Non-super reflexive space

Suppose $X$ is a reflexive space (possibly non-separable) which is not super-reflexive. Then (by definition) there exists a non-reflexive Banach space $Y$ which is non-reflexive but is finitely ...

**7**

votes

**0**answers

256 views

### Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...

**4**

votes

**1**answer

143 views

### Generator of a $C_0$-semigroup restricted to a subspace

Suppose we have a decreasing filtration of Banach spaces $(E_t)\_{t\geq0}$, inclusions $V_{s+t,t}:E_{s+t}\to E_t$ and projections $P_{t,s+t}:E_t\to E_{s+t}$ such that $P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}$. ...

**4**

votes

**0**answers

350 views

### Reflexive-saturated Banach spaces

Say that a Banach space $X$ is strongly saturated by reflexive subspaces if every closed subspace $Y\subset X$ contains a further reflexive subspace $Z\subset Y$ with $\mbox{dens }Y=\mbox{dens Z}$. If ...

**3**

votes

**1**answer

180 views

### split of s.e.s. of Banach spaces

Let $l^{\infty}$ be the Banach space of all bounded real sequences with the $sup$-norm and $c$ the closed subspace of convergent sequences. Is there a continuous linear map $T: l^{\infty} \rightarrow ...

**7**

votes

**0**answers

340 views

### Minkowski's Inequality for Integrals in Orlicz spaces

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.
Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, ...

**3**

votes

**1**answer

144 views

### Algebras with countable chains only

Is there an example of an uncountable Boolean algebra $B$ in which every chain is countable and such that $\ell_\infty$ embeds into the Banach space $C(\mbox{Stone }B)$? The latter requirement is not ...

**2**

votes

**1**answer

208 views

### Embedding of $\ell_p$ into infinite direct sums

Let $p\in (1,\infty)$ and let $q$ be conjugate to $p$. Is there a subspace of $\ell_1(\ell_p)$ isomorphic to $\ell_q$? Of course, I am uninterested in the case $p=2$.

**4**

votes

**1**answer

197 views

### Gaussian Valued Random Variables in Geometry of Banach Spaces

Why are Gaussian valued random variables so important in the Geometry of Banach spaces? I am reading the monograph by Pisier - "Probabilistic Methods in the Geometry of Banach Spaces" and in the very ...

**3**

votes

**1**answer

198 views

### Pitt's theorem for non-separable $\ell_p$ spaces

A short variant of Pitt's theorem is the followig: for $1\leq p < r <\infty$ holds
$$
\mathcal{B}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))=\mathcal{K}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))
$$
Now ...

**6**

votes

**2**answers

336 views

### Ultrapowers of operators

Can we prove that for each infinite dimensional Banach space $X$ and any free ultrafilter (possibly over uncountable set of indices) $\mathcal{U}$ the obvious embedding
...

**0**

votes

**2**answers

172 views

### ANR Subsets of banach spaces

I need a reference for conditions on a closed subspace of a Banach space to have the homotopy type of an ANR.

**0**

votes

**1**answer

150 views

### Complemented subspaces of $\ell_p(I)$ for uncountable $I$

I was looking for an article mimicing result of Pelczynski for $\ell_p$. I have found this one
Rodriguez-Salinas, B. (1994). On the Complemented Subspaces of $c_0(I)$ and $\ell_p(I)$ for $1 < p ...

**10**

votes

**3**answers

1k views

### Projections in Banach spaces

Dear All,
I am absolutely lost in the following problem:
Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm ...

**2**

votes

**0**answers

82 views

### Form of finite dimensional contractive projection in $L_p$

Let $P$ be a finite dimensional contractive (norm 1) projection in $L_p$, $1 < p < \infty$. Then $P$ is of the following form:
$Pf = \sum_{k=1}^n g_k \int h_kf$
Where $\|g_k\|_p = \|h_k\|_q = ...

**2**

votes

**0**answers

142 views

### Density of adjoint operators

I am interested in operators on a non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{\ast\ast})$ by ...

**3**

votes

**1**answer

262 views

### Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$

I'm looking for articles describing or proving nonexistence of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$.
Since $\ell_q^m$ is finite ...

**7**

votes

**2**answers

767 views

### Direct proof of injectivity of $L_\infty$

I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result.
$L_\infty$ as ...

**0**

votes

**0**answers

117 views

### $\ell_\infty^*$ has Dunford--Pettis property

hi, I'm trying to prove that $\ell_\infty^*$ has the Dunford–Pettis property. It's enough to show that $\ell_\infty$ does not contain a copy of $\ell_1$ … but I'm having some trouble doing that. Can ...

**3**

votes

**0**answers

131 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?

**2**

votes

**2**answers

314 views

### Weak topology of WOT

Let $E$ be a reflexive Banach space and let $B(E)$ be the space of bounded operators on $E$ endowed with the weak operator topology. In particular, the unit ball of $B(E)$ is then WOT-compact. $(B(E), ...

**8**

votes

**2**answers

261 views

### Projections onto $n$-codimensional subspaces of a Banach space: norms.

Hello, I'd like some help to find an answer I've been looking for since this morning.
Let $X$ be a Banach space and let $Y$ be an $n$-codimensional subspace of $X$. Let $P$ be a projection from $X$ ...

**2**

votes

**1**answer

179 views

### BM-distances between $B(\ell_p^n)$ and $\ell^{n^2}_p$

Let $\ell_p^n$ be the $n$-dimensional real or complex $\ell_p$-space and let $\mathscr{B}(\ell_p^n)$ be the space of matrices on $\ell_p^n$ endowed with the operator norm. I am looking for any ...

**9**

votes

**1**answer

318 views

### Inequivalent complete norms and the axiom of choice

Hi,
I've been wondering about the following :
Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space?
All the examples of inequivalent complete norms ...

**2**

votes

**1**answer

189 views

### Subalgebras of $B(E)$

Let me fix an infinite-dimensional (complex) Banach space $E$. There is a cute result of Bracic and Kuzma which says that every maximal abelian subalgebra of $\mathscr{B}(E)$, the algebra of bounded ...

**1**

vote

**1**answer

208 views

### Reflexive Besov spaces

I don't know whether the Besov space $B^1_{1,1}$ on a one dimensional torus is reflexive or not? Can someone help me please?

**10**

votes

**1**answer

647 views

### Non-probabilistic proof of the Johnson–Lindenstrauss lemma

The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...

**0**

votes

**1**answer

325 views

### Double duals characteristic [closed]

Recall that (for $1\le p<\infty$), $\ell^p = \{\{a_n\}_{n=1}^\infty:\sum\limits_{i=1}^\infty|a_i|\lt\infty\}$, with norm $||\{a_n\}||=(\sum\limits_{i=1}^\infty|a_i|^p)^{\frac{1}{p}}$.
It is well ...

**1**

vote

**3**answers

239 views

### Extension of lipschitz functions along a curve

Given a curve $\gamma$ in a Banach space $X$ and a function f defined along the curve s.t.
$$\big\Vert f(\gamma(t))-f(\gamma(s))\big\Vert\\leq L\big\Vert\gamma(t)-\gamma(s)\big\Vert$$
is it possible ...

**0**

votes

**0**answers

223 views

### Convergence of a function in a metric space to its metric.

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric:
If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...

**5**

votes

**1**answer

327 views

### Complemented Subspaces and Riesz-Thorin interpolation

Riesz-Thorin interpolation may sometimes be applied to subspaces (of $\ell^p$ or $L^p$) when these are complemented and the spaces in the complementation comes from a common dense subspace. To be a ...

**0**

votes

**1**answer

479 views

### Infinite linear span vs closed linear span

Hi,
Suppose we have a (real, separable) Banach space $V$ and a (linear) set $A\subseteq V$. I presume in general it might not be possible to write every element of the closed span of $A$ as an ...

**2**

votes

**2**answers

455 views

### How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional Lp-normed vector space?

Say you have a finite-dimensional vector space $V$ with an $L^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $L^p$ norm, so the unit sphere in $V_s$ ...

**3**

votes

**1**answer

444 views

### Around Banach isomorphism theorem

Let $E$ be a normed (real) space which is not complete. Is it always possible to find $f$ a continuous bijective linear function from $E$ to $E$ such that $f^{-1}$ is not continuous?

**4**

votes

**1**answer

155 views

### Set of unitaries with “spread-like” properties

I'm interested in finding two sets of $N$ unitary $N \times N$ matrices $U_{1}, \ldots, U_{N}$, $V_{1}, \ldots, V_{N}$ such that:
$
\sup\limits_{X, Y}\sum\limits_{j,k = 1}^{N} ...

**24**

votes

**3**answers

1k views

### Surjectivity of operators on $\ell^\infty$

Can anyone give me an example of an bounded and linear operator $T:\ell^\infty\to \ell^\infty$ (the space of bounded sequences with the usual sup-norm), such that T has dense range, but is not ...

**1**

vote

**1**answer

135 views

### How (and when) to factor a function defined on a product of metric spaces?

Suppose we have a set of regular functions defined on a product of metric spaces, for instance the Banach space of the smooth functions from $\mathbb R^n$ to $\mathbb C$. We know, thanks to the Taylor ...

**7**

votes

**0**answers

453 views

### Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...

**3**

votes

**1**answer

295 views

### Separable $L_1$-predual

Some isometric preduals of $\ell_1$ are of the form $C_0(K)$ where $K$ is countable. I am wondering whether this is a general rule.
Question: Is there a measure $\mu$ and a (preferably separable) ...

**8**

votes

**0**answers

244 views

### Preduals of $\ell_1$

The space $\ell_1$ has loads of (isomorphic) predulas. They can be as weird as possible but I am interested in Banach lattices.
Question: Let $X$ be a Banach lattice with dual isomorphic to $\ell_1$. ...

**6**

votes

**2**answers

652 views

### Weak*-closed and complemented subspaces of dual Banach spaces

We consider a Banach space $X$ and its dual $X^*$.
Let $Q\colon X^\ast \to X^\ast$ be an idempotent operator.
Question:
Can we find an idempotent operator $P\colon X^\ast \to X^\ast$ which is ...

**6**

votes

**0**answers

232 views

### What is the intersection of the closures of left invertible operators and right invertible operators?

From Douglas Zare's answer (see Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?), one know that
$$ \overline{G_{l}(X,Y)} \bigcap \overline{G_{r}(X,Y) } = ...

**13**

votes

**2**answers

659 views

### Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?

Let $X$ and $Y$ be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that $X$ embed in $Y$, and write $X \preceq Y$, if there exists a left invertible operator ...

**14**

votes

**3**answers

604 views

### A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...

**7**

votes

**0**answers

502 views

### Strictly singular operators and their adjoints

This is a question I thought about a while back and figured I'd throw it out there to see if anyone has some insight that I am missing.
Let $X$ and $Y$ be infinite dimensional separable Banach ...

**1**

vote

**0**answers

175 views

### When Pelczynski's property (V*) forces (V) in the dual space?

Recently, I have been asked if are there any sufficient conditions for a Banach space with Pelczynski property (V*) to have dual with property (V). Since it was a long standing open problem, I guess ...

**0**

votes

**1**answer

147 views

### Reference request for sums of Grothendieck spaces

I much appreciate your help with my previous question. Now, I'd like to ask about a reference. I need a fact asserting that if $p\in (1,\infty]$ (with emphasis on the case $p=\infty$) then the ...