# Tagged Questions

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### Open Mapping Theorem: counterexample

The Open Mapping Theorem says that a linear continuous surjection between Banach spaces is an open mapping. I am writing some lecture notes on the Open Mapping Theorem. I guess it would be nice to ...
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### How to construct an “explicit” element of $(\ell^\infty(\mathbb N))^* \setminus \ell^1(\mathbb N)$? [duplicate]

Possible Duplicate: Nonnegative linear functionals over $l^\infty$ An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? Everything is in the title: How to ...
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### completeness of cones in an ordered normed space

Let $(X,\|\cdot\|,\le)$ be a normed, ordered vector space over $\mathbb{R}$ and let $X^+=\lbrace x\in X:x\ge0\rbrace$ denote the (positive) cone in $X$. with a metric $d$ induced by the norm ...
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### Convex hull of the extreme points of the unit ball in $C(K)$ where $K$ is the Cantor set.

Let $K$ denote the usual $1/3$ Cantor set and let $B=B_{C(K)}$ (here $B_{C(K)}$ = {$v \in C(K) : \|v\| \leq 1$} denotes the closed unit ball of $C(K))$. Then how to prove that $B$ coincides with the ...
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### Exhibiting open covers with no finite subcovers.

How do I exhibit an open cover of the closed unit ball of the following: (a) $X = \ell^2$ (b) $X=C[0,1]$ (c) $X= L^2[0,1]$ that has no finite subcover?
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### Example for a proper dense subspace?

I have been reading some books on functional analysis, and many of them keep talking about a vector space along with a dense proper subspace of it (especially when constructing counterexamples). But ...
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I got stuck at the following problem. Let $X,Y$ be normed spaces. A bounded linear operator $\tau\in\mathcal{B}(X,Y)$ is called strictly coisometric if $$... 3answers 2k views ### Discontinuous linear functional I'm trying to find a discontinuous linear functional into \mathbb{R} as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since \ell_2 is infinite-dimensional, ... 1answer 120 views ### Bidual of a WSC space Let E be a Banach space which is weakly sequentially complete (i.e. each weak Cauchy sequence converges weakly). Must E^{**} be weakly sequentially complete either? Of course, this question is ... 1answer 706 views ### Reflexive space which is not uniformly convex I found this beautiful theorem (Milman-Pettis): Every uniformly convex Banach space is reflexive I think it's a remarkable statement, since uniformly convexity is a geometric property of the norm ... 1answer 163 views ### Is quasinorm always continuous? Let X be a vector space over \mathbb{R} or \mathbb{C}, \|\cdot\|: X\rightarrow [0,\infty) is called a quasi-norm if i) \|x\|=0 \Rightarrow x=0 ii) \|\lambda x\|=|\lambda|\|x\|, \forall ... 1answer 449 views ### Isometric to Dual implies Hilbertable? Let X be a Banach space and suppose that X is isometric to its continuous dual space X^*. Must X be hilbertable in the sense that there exists an inner product which induces the norm on X? ... 3answers 2k views ### Is it possible for a function to be in L^p for only one p? I'm wondering if it's possible for a function to be an L^p space for only one value of p \in [1,\infty) (on either a bounded domain or an unbounded domain). One can use interpolation to show that ... 1answer 554 views ### Does separability follow from weak-* sequential separability of dual space? Let E be a Banach space. Suppose that E' is weakly-* sequentially separable, that is, that there exists a countable D \subset E' s.t. every x' \in E' is a limit point of a ... 3answers 498 views ### How is the uniform boundedness principle compatible with this seemingly weak convergent sequence? In showing that (x_i\rightharpoonup x)\not\Rightarrow(x_i\to x) or similar noncorallaries, one frequently uses the counterexample$$ (u_i)_{i\in\mathbb{N}}\in \ell^2\colon \quad u_i = ...
The method of Lagrange multipliers is used to find the extrema of $f(x)$ subject to the constraints $\vec g(x)=0$, where $x=(x_1,\dots,x_n)$ and $\vec g=(g_1,\dots,g_m)$ for $m \leq n$. Although ...