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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Biorthogonal functionals

If $X$ is a separable Banach space and $(x_n)$ is a basic sequence, then we can define biorthogonal functionals $(x^{*}_n)$ in $X^{*}$ such that $x^{*}_n(x_k)=\delta_{nk}$. What about conversely? If ...
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87 views

Trivial intersection of kernels

This is a follow up to the question: Biorthogonal functionals. A positive answer to that question implies a negative answer to this one. If $X$ is a separable Banach space, can we find a basic ...
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1answer
49 views

Dual of colimit in $\text{Ban}_1$

I learned in J. Castillo's Hitchhiker guide to categorical Banach space theory that, by a theorem of Semadeni and Zidenberg, limits and colimits exist in the category $\text{Ban}_1$ of Banach spaces ...
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Elements of Minimal Norm on an affine subset of a Banach Space

Let $v_1$ and $v_2$ be two elements of a real Banach space $(X,\lVert\;\rVert)$, each of unit norm. Consider the one-dimensional affine subspace $v_1+tv_2$ for real $t$. Is there a general formula for ...
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81 views

A single point as sum range of a series

Can anyone give some clues to show that any infinite dimensional Banach space have a conditionally convergent series whose sum range is a single point? Thanks in advance for any help.
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Extension of a function from almost everywhere to everywhere

The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points? Example: Let ...
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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 ...
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1answer
106 views

Two questions on the James $p$-space $J_{p}(1<p<\infty)$

Let $1<p<\infty$. The James $p$-space $J_{p}$ is the Banach space of all sequences of real numbers $(a_{i})_{i}\in c_{0}$ such that ...
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A question on the quantification of compact operators

Let me first fix some notations. If $A$ and $B$ are nonempty subsets of a Banach space $X$, we set $$d(A,B)=\inf\{\|a-b\|:a\in A,b\in B\},$$$$\widehat{d}(A,B)=\sup\{d(a,B):a\in A\}.$$ Let $A$ be a ...
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317 views

Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved). Then the Fourier transform of this function is given by ...
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Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?

Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
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A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again. Take the vector space of infinitely ...
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quasi-weakly compact operators, co-ideals of operator ideals, and Banach spaces $X$ with $X^{**}/X$ separable

Throughout, $X$ and $Y$ will denote Banach spaces with $T\in\mathcal{L}(X,Y)$ (the space of continuous linear operators between $X$ and $Y$). We define the operator ...
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1answer
83 views

Various limits of the Christoffel Darboux Kernel

In a different thread, we stumbled upon the following question: Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective ...
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154 views

When is the sum of complemented subspaces complemented?

Let $X$ be a Banach space. Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto ...
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Weak compactness in the James space and its dual

It is known that there are characterizations of weak compactness in most of classical non-reflexive spaces (e.g. $L_{1}$-spaces and $C(K)$-spaces). I wonder whether there are characterizations of weak ...
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Normed space between $H^{0+}$ and $L^2$

In the space $\in L^2(\mathbb{R}^3)$, consider the following condition. $$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$ Of course if $f\in H^s(\mathbb{R}^3)$ ...
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64 views

Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates an analyitc ...
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670 views

Who proved that $l^1$ and $L^1[0,1]$ are not isomorphic?

$l^1$ has the Schur property (every weakly convergent sequence is norm convergent) and $L^1[0,1]$ does not, so the two spaces cannot be isomorphic. Is this folklore, or is it credited to someone? ...
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101 views

Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
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How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional $\ell^p$-normed vector space?

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

Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces

The Mazur-Ulam theorem (1932) states that any isometry of a normed linear space is affine. See Nica (Expo. Math. 30 (2012), 397-398; arXiv:1306.2380) for a very elegant proof. Question: Let $M$ be a ...
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Good reference for noncommutative $L^p$ spaces

I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...
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255 views

Compact non-nuclear operators

I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, ...
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135 views

Constructing Extreme Points in Reflexive Banach Spaces

A theorem of Lindenstrauss and Phelps states that if $X$ is a separable reflexive Banach space then the unit ball of $X$, $Ba(X)$, has uncountably many extreme points. The proof goes by contradiction ...
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Attempted Banachification of a space

In a recent blog post, Terry Tao mentioned the question of how to tell if a Hausdorff topological vector space admits a finer topological structure which happens to be the topology of a Banach space ...
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On the weakly sequential completeness of the dual of the James space $J$

Let me first introduce some definitions. Let $1\leq p\leq \infty$. A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be weakly $p$-convergent to $x\in X$ if the sequence $(x_{n}-x)_{n}$ is ...
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$l^1$ versus $l^2$

Is there an elementary proof of this Banach space fact? If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite ...
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Unconditionally $p$-converging operators on $L_{1}[0,1]$

Let $1\leq p<\infty$. We say that an operator $T:X\rightarrow Y$ is unconditionally $p$-converging if $T$ takes a weakly $p$-summable sequence to a norm null sequence. Question: Is every ...
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The dual space of $C[0,1]$

Since $C[0,1]^{*}$ is an abstract $L$-space, $C[0,1]^{*}$ is order isometric to $L_{1}(\mu)$ for some measure $\mu$. My question is: the measure $\mu$ can be choosen to be a finite positive measure? ...
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Do non-stable Banach spaces exist?

Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties: Is every infinite ...
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Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space

I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces". We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...
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Classification of subsymmetric basic sequences

I just encountered the following statement: any subsymmetric basic sequence is either weakly null or equivalent to the unit vector basis of $\ell_1$. It's the first time I realize that. I do see the ...
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Maurey-Pisier Theorem for complex Banach spaces

A famous theorem of Maurey and Pisier, usually stated for real Banach spaces, says that $\ell_a$ and $\ell_b$ are finitely representable in $X$, where $a$ is the supremum of the $p$ such that $X$ has ...
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290 views

Invariant probability on a unit ball of a Banach space

Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries. Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) ...
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A Banach space with the BD property and without the weak Gelfand-Phillips property

A subset A of X is called Grothendieck if every operator T from X to $c_0$ maps A to a relatively weakly compact set. A Banach space has the weak Gelfand-Phillips property (wGP) if every ...
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Extracting a subsequence for which $\sup_j \left\vert \text{supp}(x_{n_j}) \right\vert < \infty$

Let $X$ be a Banach space with basis $(e_n)_{n=1}^\infty$, and suppose that $(x_i)_{i=1}^\infty$ is a normalized block basic sequence of $(e_n)_{n=1}^\infty$. In addition assume that ...
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179 views

Non-equivalence of admitting different types of bases in Banach spaces

Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis ...
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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 ...
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A question on measuring weak non-compactness in $L_{1}(\mu)$ spaces

Let me first fix some notations. Let $X=L_{1}(\mu)$, where $\mu$ is a finite positive measure. Let $A$ be a bounded subset of $X$. Set $\omega(A)=\inf\{\widehat{d}(A,K):\emptyset\neq K\subset X$ is ...
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Where can I find some articles and lecture notes in renorming theory in Banach spaces? [closed]

I am really into renorming theory in Banach spaces especially, renorming in non-reflexive Banach spaces such that they have nice property, for example they have fixed point property,locally uniformly ...
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Kolmogorov width for cartesian products

For an operator $T:X\to Y$ between Banach spaces with unit balls $B_X$ and $B_Y$ the sequence of Kolmogorov widths is $$ \delta_n(T)=\inf\lbrace \delta>0: T(B_X)\subseteq \delta B_Y +L \text{ for ...
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Existence of strictly convex Banach space without weak normal structure

This is a complementary question to the following at Existence of normal structure in strictly convex Banach spaces Does there exist a strictly convex Banach space without the weak normal structure ...
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1answer
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Is the topological dual of a Banach space weakly* closed in its algebraic dual?

The question is completely contained in the title :) I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential ...
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1answer
196 views

Banach-Stone Theorem in Lipschitz-free spaces

If $T$ is a nonlinear surjective isometry from Lipschitz-free space $\mathcal{F}(M)$ to $\mathcal{F}(N)$($M,N$ are metric spaces), is $M$ homeomorphic to $N$?
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Decomposable Banach Spaces

An infinite dimensional Banach space $X$ is decomposable provided $X$ is the direct sum of two closed infinite dimensional subspaces; equivalently, if there is a bounded linear idempotent operator on ...
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1answer
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Spectrum of compact operator between different Banach spaces

Let $X, Y$ be two different Banach spaces, and let $T: X \to Y$ be a compact linear operator. Suppose the identity $I : X \to Y$ is well-defined. (For example, we could have $X = L^2([0,1])$ and $Y = ...
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Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $||x|| = 1$ and $||Ax|| = ||A||$. The ...
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$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$

$c_0(C[0,1])$ is the $c_0$-direct sum of countably many $C[0,1]$.How to prove $C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$. Here,Banach-space isomorphism means a bounded invertible operator ...
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Second duals of Grothendieck spaces

The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{(4)}$, $\dots$ are Grothendieck spaces. (See, e.g., this note ...