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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Renorming a Banach space to make projections contractive

Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$. Can the same be done for a family of projections? That is, given finitely many ...
3
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1answer
87 views

almost invariant half space for a dual of a restricted operator

Let $X$ be an infinite-dimensional Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. A closed subspace $Y$ of $X$ is said to be an almost-invariant halfspace (...
2
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67 views

Self-adjoint, strictly singular operators on Hilbert spaces

Let $X$ and $Y$ be infinite-dimensional Banach spaces. Recall that an operator $T: X\rightarrow Y$ is strictly singular if it is not an isomorphic embedding when restricted to any infinite-dimensional ...
2
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78 views

Products of strictly singular operators on $L_{p}[0,1]$ or on $C[0,1]$

In 1970, V.D. Milman (Operators of class $C_{0}$ and $C^{*}_{0}$, Teor. Funkc. Funkc. Anal. Ih Priloz. 10(1970),15-26) proved that the product of two strictly singular operators on $L_{p}[0,1](1\leq p&...
2
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171 views

A unital algebra with norm and continuous multiplication is a Banach algebra

In my research in functional analysis, I came across this rather simple result: For a normed algebra A over $ \mathbb{C} $ with unit, in which multiplication , right and left are both continuous w....
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721 views

Meager subspaces of a Banach space and weak-* convergence

I previously asked a version of this question on Math.SE, but didn't receive an answer. (But there is a bounty there if you want to claim it!) Let $X$ be a Banach space. (If it helps, feel free to ...
8
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1answer
239 views

Interpolation between $L_1^0$ and $L_2^0$

Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...
2
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322 views

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 ...
3
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181 views

On the Lorentz sequence space $d(w,1)$

I am interested in examples of dual Banach spaces $X$ with the Schur property (weakly convergent sequences in $X$ are norm convergent) like $\ell_1$. The Lorentz spaces $d(w,1)$ [Lindenstrauss and ...
4
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410 views

Cameron Martin space

I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated. 1) It is the ...
5
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63 views

Set of w*-continuous operators closed for the weak* topology or not?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
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96 views

quasi-nilpotent part of a dual operator

Definitions and notation. Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as \begin{equation*}H_0(T):=\left\{...
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40 views

A singleton as domain sum of a series [migrated]

Consider the series $$e_1+\frac{1}{2}e_2-\frac{1}{2}e_2+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{8}e_4-\frac{1}{8}e_4+\cdots-\frac{1}{8}e_4+\frac{1}{16}e_5-\cdots$$ in the ...
7
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1answer
335 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 $...
7
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1answer
86 views

About $C(K)$-spaces containing no copy of $l_{1}$

Let $K$ be a compact Hausdorff space. I wonder whether there are characterizations of $K$ such that $C(K)$ contains no copy of $l_{1}$. There are some compact Hausdorff spaces $K$ such that $C(K)$ ...
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74 views

Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$

Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...
8
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180 views

Davis, Figiel, Johnson and Pełczyński factorization through spaces with a bases

Davis, Figiel, Johnson and Pełczyński's Factorization Theorem states that each weakly compact operator $T:X \to Y$ between Banach spaces $X$ and $Y$ factors through a reflexive Banach space $Z$. In ...
0
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1answer
208 views

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 $f\in H^s(\mathbb{R}^3)$ ...
0
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1answer
78 views

Sequentially continuous but not continuous linear map $(X^*, w^*)$ to $(Y^*,w^*)$

Let $X, Y$ be Banach spaces and let $T : (X^*, w^*) \rightarrow (Y^*,w^*)$ be a linear map. Suppose that $T$ is sequentially continuous. Must $T$ be weak*-to-weak*-continuous ?
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76 views

Subprojectivity of the spaces $B_{p}(1<p<\infty)$

Fix $1<p<\infty$. In his thesis, C.J.Seifert adapted the construction of Baerstein's space $B$ by replacing the number 2 by $p$ to construct the space, called "$B_{p}$" space. He showed that ...
1
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1answer
62 views

Subprojectivity of $L_{p}(p>2)$

Let $p>2$. Following from M.I. Kadec and A. Pełczyński's results (Studia Math. 1962), R.J.Whitley (Trans. Amer. Math.Soc. 1964) observed that $L_{p}$ is subprojective, that is, every infinite-...
3
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102 views

On the complemented subspaces of $L_{p}(p>2)$

M.I. Kadec and A. Pełczyński proved that if $E$ is a subspace of $L_{p}(p>2)$ isomorphic to $l_{2}$, then $E$ is complemented in $L_{p}$. My question is: Is there a constant $C_{p}$ depending only ...
2
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71 views

Reflexive subspaces of dual spaces

If $X$ is a Banach space, is it true that $X^{*}$ must contain an infinite dimensional reflexive subspace? I know of Gowers' example of a Banach space not containing $c_0$, $l_1$, or reflexive, but I ...
5
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147 views

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 ...
4
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111 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 ...
3
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1answer
54 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 ...
2
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1answer
102 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.
2
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185 views

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 $f(x)=\...
8
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218 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 ...
2
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1answer
110 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 $$\|(a_{i})_{i}\|=\sup\{(\sum_{j=1}^{n}|a_{p_{j-1}}-a_{p_{j}}|...
3
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1answer
100 views

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 ...
23
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1answer
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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?
3
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287 views

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 ...
5
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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 $\overline{T}\in\mathcal{L}(X^{**}/...
1
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1answer
86 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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159 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 ...
0
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1answer
67 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 ...
20
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1answer
707 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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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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723 views

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 $V_s$...
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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 ...
4
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101 views

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 ...
4
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1answer
260 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, non-...
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143 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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382 views

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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77 views

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 ...
8
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311 views

$l^1$ 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 ...