Questions tagged [banach-spaces]

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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3 votes
1 answer
198 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 $X$. ...
29 votes
6 answers
8k views

Nonseparable Hilbert spaces

Being nonseparable Banach space is in fact nothing special: one meets the first examples in the standard functional analysis course, when one learns about $\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
12 votes
3 answers
14k 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$(c_0)$? EDIT: As confirmed in the comments, the OP intended to ...
4 votes
0 answers
165 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 ...
4 votes
1 answer
503 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; ...
6 votes
2 answers
192 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 ...
9 votes
1 answer
346 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 ...
3 votes
1 answer
139 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 \...
5 votes
1 answer
940 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 ...
1 vote
1 answer
90 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}...
1 vote
1 answer
181 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-...
3 votes
1 answer
160 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 ...
2 votes
0 answers
328 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 $...
0 votes
1 answer
116 views

A characterization of reflexivity of Banach spaces via convex block sequences

By James's Theorem, A. Ulger (Weak compactness in $L^{1}(\mu.X)$, Proc. Amer. Math. Soc. 113(1991),143-149.) proved that a bounded subset $A$ of a Banach space $X$ is relatively weakly compact if and ...
4 votes
1 answer
259 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* ...
2 votes
1 answer
206 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^{*...
1 vote
0 answers
728 views

A weakly sequentially continuous operator which is not weakly continuous

I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity. So, let $T$ an operator between a Banach space $X$ and itself. $T$ is weakly ...
2 votes
0 answers
129 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_{\...
6 votes
1 answer
204 views

$C[0,1]$ fails the property (K)

Recall that a Banach space $X$ has the property (K) if every $w^{*}$-convergent sequence in $X^{*}$ admits a convex block subsequence which converges with respect to the Mackey topology. The property (...
3 votes
1 answer
222 views

Measurable selection for argmin to distance

Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with ...
0 votes
0 answers
368 views

Weak topology on spaces of measures and Borel sets

Let $K$ be a compact Hausdorff space (not necessarily metric or even separable). Let $M(K)$ be the space of all Radon measures on $K$ (that is, finite signed regular Borel measures) endowed with the ...
1 vote
0 answers
72 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^{...
3 votes
1 answer
171 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 ...
2 votes
0 answers
37 views

Weak relaxation of a strongly lower semi-continuous functional

Let $F$ be a lower semicontinuous functional on a Banach space $X$, wrt its strong topology. Is there a known form for the relaxation (lower semicontinuous envelope) of $F$ with respect to the weak ...
4 votes
0 answers
144 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 ...
8 votes
0 answers
183 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 ...
4 votes
0 answers
74 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 ...
6 votes
0 answers
83 views

Finding approximate eigenvectors: quantitative results

Let $X$ be a complex Banach space and $T \colon X \to X$ be a bounded operator. For every $x \in X \setminus \{0\}$, denote by $Y_x$ the smallest closed $T$-invariant subspace of $X$ containing $x$. ...
11 votes
1 answer
1k views

Dual of the space of all bounded holomorphic functions

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
19 votes
6 answers
8k views

Unbounded operator bounded in a dense subset

Let $X, Y$ be normed vector spaces, where $X$ is infinite dimensional. Does there exist a linear map $T : X \rightarrow Y$ and a subset $D$ of $X$ such that $D$ is dense in $X$, $T$ is bounded in $D$ (...
5 votes
1 answer
145 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^\...
5 votes
1 answer
149 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 ...
2 votes
1 answer
186 views

Are there minimal topological conditions on a space 𝑋 for it to have a countable separating set?

Are there minimal topological conditions on a space $X$ for it to have a countable separating set? A separating set here is a set $D \subset C(X)$ (where $C(X)$ is the space of continuous functions ...
2 votes
1 answer
68 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 ...
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 ...
2 votes
0 answers
62 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(\...
3 votes
1 answer
136 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 $r>s>2$? ...
1 vote
0 answers
111 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 ...
0 votes
1 answer
75 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,\...
5 votes
0 answers
422 views

A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?

In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
1 vote
0 answers
63 views

Connected components of bounded linear operators of $V = (\mathcal C(U(1), \mathbb C) , \lVert \cdot \rVert_\infty)$

This question is related to this one. Consider the complex Banach space $V=(\mathcal C(U(1), \mathbb C), \Vert \cdot \Vert_\infty)$ where $\mathcal C(U(1), \mathbb C)$ is the space of continuous ...
4 votes
1 answer
274 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 ...
3 votes
0 answers
115 views

With Khinchine's inequality, prove Fourier basis is unconditional in $L^{p}[0,1]$ only for $p=2$

I am trying to prove Problem 6.10 in "Classical and Multilinear Harmonic Analysis" by by Camil Muscalu and Wilhelm Schlag. Problem Problem 6.10. Let $1\le p < \infty$ and suppose that ...
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-...
5 votes
1 answer
371 views

A subcontinuous function, which is not continuous

Let $E$ be a Banach space and $T: E\rightarrow E$ be a mapping. $T$ is said to be subcontinuous if for any sequences $(u_n)_{n\in\mathbb{N}}$ in $E$ that converge strongly to $u$ the sequence $(T(u_n)...
2 votes
0 answers
959 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)|...
1 vote
1 answer
184 views

On examples of subspaces of $C(X)$ for which state spaces are Choquet Simplices

Let $C(X)$ be the Banach space of all Real valued continuous functions on a compact Hausdorff space $X$. What are examples of uniformly closed subspace $\mathcal{A}$ of $C(X)$ such that $\mathcal{A}$ ...
9 votes
1 answer
231 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 ...
2 votes
0 answers
125 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 ...
4 votes
1 answer
165 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 ...

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