4
votes
0answers
57 views

Non-reflexive Orlicz spaces

I am looking for a good reference on Orlicz spaces; I would appreciate any books treating this topic from the Banach-space perspective. For example, I would like to find a reference to the following ...
3
votes
1answer
320 views

A useful criterion in vector integration

I would like to know the proof of the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let ...
5
votes
0answers
192 views

On the classification of injective Banach spaces

A Banach space $E$ is injective when it is complemented in each Banach space $X$ that contains it as a closed subspace. The space $E$ is $1$-injective if the copy of $E$ in $X$ is the range of a ...
14
votes
0answers
426 views

An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
1
vote
0answers
71 views

Maximum Principle with Banach Control Space

This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in ...
3
votes
1answer
208 views

Orthonormal basis in $\ell^n_p$

Given a $k$-dimensional subspace in $\ell^n_p$, is there a way to bound the value of $$ \sum_{i=1}^k \|a_i\|_{\ell^p}^2 $$ for $a_i$ an orthonormal (for the "standard" underlying $\ell^n_2$) basis. ...
1
vote
0answers
83 views

Volume Function on Banach Spaces

I'm looking for a reference for the following so-called Volume Function $V_n$, which is intended to be a Banach/normed vector space generalization of the determinant. Let $X$ be a Banach space with ...
3
votes
0answers
66 views

$x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology

This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...
1
vote
1answer
125 views

Special kind of operators

Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation $$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$ where ...
4
votes
1answer
138 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 $P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}$. ...
2
votes
0answers
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 = ...
3
votes
1answer
256 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 ...
2
votes
1answer
178 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 ...
1
vote
3answers
237 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 ...
1
vote
1answer
133 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
0answers
436 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 ...
0
votes
1answer
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 ...
3
votes
1answer
344 views

Determining continuous functions on Banach spaces

Let $X$ be a real Banach space. For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we┬┤ll say that $\mathcal{F}$ determines $g$ if ...
2
votes
1answer
225 views

Generalized unique nearest point problem for a compact, convex set in a strictly convex Banach space.

If $X$ is a Real Banach space with strictly convex norm, it is known that for any non-empty compact, convex set $K$ and point $x_0\notin K$, there exists a unique point $z_0\in K$ minimizing the ...
6
votes
2answers
488 views

Continuous choice of Hahn-Banach extensions

QUESTION RETRACTED - My original argument was fundamentally mistaken (mixing up lower and upper semi-continuity). Sorry (and thanks for the useful comments) I need, and (unless I am seriously ...
13
votes
1answer
367 views

Is the space of Hankel operators complemented in B(H)?

Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$. Let ...
7
votes
3answers
631 views

Preduals of B(E)

For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators. If $E$ is a Banach space then is it known whether $B(E)$ is always ...
6
votes
1answer
274 views

Best constant in comparison between Rademacher and gaussian averages?

Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables. What is the best ...
8
votes
1answer
662 views

Morphisms of Banach spaces

What is the standard name in English for bounded linear maps $f:E\to F$ between Banach spaces such that the kernel $\ker(f)$ has a complement, and $\text{im}(f)$ is closed, and has a complement? ...
12
votes
0answers
1k views

Infinite convex combinations in a Banach space

Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds: For any sequence $(x_k)_{k\ge0}$ in $C$, and for any sequence of non-negative real numbers ...
5
votes
1answer
581 views

Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$

Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...
7
votes
2answers
1k views

Interpolation of Sobolev spaces

I know quite a bit about the abstract theory of Interpolation of Banach spaces. Today I had a colleague from Environmental sciences (who used to be in our Applied Maths department) come and ask me ...
0
votes
0answers
271 views

Lifting of product of a Banach algebra

Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product. A lifting of $T$ is ...
11
votes
2answers
940 views

“Orthogonal complement” of a subspace of a Banach space

I'm looking for a reference. I am considering the situation where $X$ is a Banach space and $Y$ is a closed finite co-dimensional subspace. I am looking for a $W$ that is a complement of $Y$ (i.e. so ...
7
votes
3answers
632 views

Non-Borel subspace of Banach space

Let $X$ be a separable Banach space, $M \subset X$ a linear subspace. Must $M$ be a Borel set in $X$? I believe the answer is "no," since I have seen authors who are careful to talk about "Borel ...
6
votes
1answer
363 views

The typical size of a random element in a Banach space

Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...
8
votes
1answer
575 views

Nonseparable Hilbert spaces as quotients of spaces of bounded functions

Is the following result true: the Hilbert space $\ell^{2}\left(2^{\Gamma}\right)$ is a quotient of $\ell^{\infty}\left(\Gamma\right)$ for any uncountable $\Gamma$ ? [I think it is, but cannot remember ...
3
votes
3answers
1k views

Examples of Banach spaces and their duals

There are many representation theorems which state that the dual space of a Banach space $X$ has a particularly concrete form. For example, if $X = C([0,1],\mathbb R)$ is the space of real-valued ...