# Tagged Questions

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

215 views

344 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 ...
52 views

### Bessaga-Pelczynski Selection Principle

Bessaga-Pelczynski Selection Principle states that if $(x_{n})_{n}$ is a basis for a Banach space $X$, then every normalized weakly null sequence $(y_{n})_{n}$ in $X$ admits a subsequence that is ...
103 views

### Choice and the Baire property in non-separable complete metric spaces

It's known to be consistent with ZF+DC that every subset of $\mathbb{R}$ has the Baire property (BP). (E.g. Shelah's model). If so, then every subset of every complete separable metric space has ...
The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix. For complex numbers $a_1,\ldots,a_n$, I will use the notation $$\mbox{... 1answer 380 views ### Dominated convergence to characteristic function Let \phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]} be the m -times convolution (so m+1 characteristic functions are involved). Then the Fourier transform of this function is given by ... 0answers 93 views ### Banach spaces complemented in their ultrapowers By the principle of local reflexivity, the second dual X^{**} of a Banach space X is complemented in some ultrapower X^U of X. Even when X is separable, the index set of U cannot be ... 1answer 68 views ### Complemented subspaces in the James space Let J be the James space. I have the following questions: Question 1: Does every infinite-dimensional closed subspace of J contain an infinite-dimensional closed subspace that is C-complemented ... 1answer 217 views ### A unital algebra with norm and continuous multiplication is a Banach algebra In my research in functional analysis, I came across this rather simple result: For a normed algebra A over  \mathbb{C}  with unit, in which multiplication , right and left are both continuous w.... 2answers 370 views ### Extracting subsequences in Banach spaces, along an ultrafilter? There are various principles in Banach space theory that allow one to pass from a given sequence of vectors (x_n), to a subsequence (x_{n_k}) with some desired property. I'm thinking here, in ... 0answers 124 views ### Norm of projection onto functions of mean zero Let X be a finite set and consider the space \ell^2(X;Y) of functions \zeta:X\to Y, where Y is a fixed Banach space. It decomposes into a direct sum of constant function and its complement \... 1answer 354 views ### Is the L^1-space dual to a Banach space Let (\Omega,\mu) be a measure space. It is well known that for 1<p\leq \infty one has the duality$$L^p=(L^{p*})^*,$$where 1/p+1/p^*=1. Question. Is it known that the Banach space L^1 is ... 1answer 234 views ### Normed space between H^{0+} and L^2 In the space \in L^2(\mathbb{R}^3), consider the following condition.$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$Of course if f\in H^s(\mathbb{R}^3) ... 0answers 70 views ### Weak Topology and Domain Theory: Which topology on the function domain restricts to the weak topology on C([0,1])? Let \mathbb{IR} be the interval domain over the set \mathbb{R} of real numbers, defined by:$$\mathbb{IR} := \{ [a,b] \mid a, b \in \mathbb{R}, a \leq b\} \cup \{ \mathbb{R}\},$$and ordered by ... 1answer 78 views ### Sequences in L_{p}(1<p<\infty) that is equivalent to the unit vector basis of l_{p} or l_{2} Let 1<p<\infty. Johnson and Schechtman (Multiplication operators on L(L_{p}) and l_{p}-strictly singular operators, 2008, DOI: 10.4171/JEMS/141, eudml, arxiv) observed that if (x_{n})_{n}... 0answers 101 views ### A universal operator between separable Banach spaces The Banach space C[0,1] is universal for all separable Banach spaces in the sense that for a separable Banach space X there is an isometric isomorphism from X into C[0,1]. My question is ... 1answer 116 views ### Subspaces of L_{p}(2<p<\infty) Let p>2 and X a subspace of L_{p}. Then Kadec and Pelczynski proved that either X is isomorphic to l_{2} or X contains a subspace isomorphic to l_{p}. Question: if X is ... 1answer 50 views ### Weakly null sequences in L_{p}(1<p<2) Let 1<p<2. Let (f_{n})_{n} be a normalized weakly null sequence in L_{p} such that the sequence (f_{n})_{n} contains no subsequence that is equivalent to the unit vector basis of l_{p}.... 1answer 139 views ### Reference: Finsler Derivative? On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ... 0answers 150 views ### Banach spaces: A ball being a subset of the interior of the union of two balls Let X be a separable reflexive Banach space and let A, B_1, and B_2 be three closed balls in X. Is there a `handy' necessary and sufficient condition for checking whether A \subseteq (B_1 \... 0answers 79 views ### Equivalence of questions regarding restrictions of pure states In Davidson and Szarek's article "Local Operator Theory, Random Matrices and Banach Spaces" in the Handbook of the Geometry of Banach Spaces, the authors discuss the (now solved) Kadison-Singer ... 1answer 82 views ### When do two quasi-Banach spaces with identical dual spaces have equivalent norms? Let X and Y be two quasi-Banach spaces such that the dual spaces satisfy X^*=Y^*. I want to know if there are some conditions that imply X=Y (in the sense of equivalent norms). 1answer 348 views ### 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 ... 0answers 69 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 ... 0answers 81 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&... 2answers 735 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 ... 1answer 258 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 ... 2answers 183 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 ... 1answer 419 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 ... 0answers 70 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_*)-... 0answers 99 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\{... 1answer 92 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) ... 0answers 112 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 ... 1answer 198 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 ... 1answer 90 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 ? 0answers 80 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 ... 1answer 66 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-... 1answer 107 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 ... 0answers 76 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 ... 2answers 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 ... 1answer 113 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 ... 1answer 55 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 ... 1answer 104 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. 1answer 187 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)=\... 1answer 220 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 ... 1answer 114 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}}|...
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 ...