Articles of functional analysis

Convergence in distribution (weak convergence)

Let $X_n$ and $X$ be random variables taking values in the metric space $(S,d)$. The sequence $(X_n)_n$ is convergent to $X$ in distribution (or weakly) if $E[f(X_n)] \to E[f(X)]$ for all $f:S\to R$ continuous and bounded. I read somewhere that it’s equivalent to consider only uniformly continuous and bounded $f$. Could you give me a […]

Hahn-Banach theorem (second geometric form) exercise

Let $X$ be a vector normed space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that $$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F).$$ Apply the Hahn-Banach theorem (second geometric form)in order to prove that there exists scalars $\alpha_1,\alpha_2,\ldots,\alpha_N$ such that $$F\ =\ \sum_{i=1}^N\alpha_iF_i.$$ Explain how do you can simplify the above proof, without Hahn-Banach (any form), but using the orthogonality, […]

Confusion in Gelfand theorem in C*-algebra.

I am reading HX Lin’s book, named “An introduction to the classification of amenable C*-algebras”, I can not understand a corollary of Gelfand theorem(Corollary 1.3.6): If a is a normal element in a unital C*-algebra A, then there is an isometric *-isomorphism from $C^{\ast}(a)$ to $C_{0}(sp(a)\setminus\{0\})$, which sends $a$ to the identity function on $sp(a)$. […]

Riesz's Lemma for $l^\infty$ and $\alpha = 1$

Riesz’s Lemma says the following: Let $X$ be a normed vector space and $Y$ a proper closed subspace of $X$. Pick $\alpha \in (0,1)$. Then $\exists x\in X$ such that $|x|=1$ and $d(x,y) \geq \alpha$ for all $y \in Y$. I am investigating why we cannot take $\alpha=1$ in general. Wikipedia says “the space $l^\infty$ […]

Dimension for a closed subspace of $C$.

Let $X \subset C^1[0,1]$ be a closed subspace of $C[0,1]$ (with sup norm). Prove that $X$ has to be finite-dimensional.

Is the result true when the valuation is trivial and $\dim(X)=n$?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are equivalent (i.e., they induce the same topology) iff there are constants $c_1$ and $c_2$ such that $c_1p_1\leq p_2\leq c_2p_1$ And here Eric Wofsey showed, […]

What are the restrictions on the covariance matrix of a nonnegative multivariate distribution.

This question is a step in answering this question on the Given a distribution $F(X_1,\ldots,X_n)$ on the nonnegative orthant $\mathbb{R}_+^n$ (i.e. each of the marginals is supported on the nonnegative reals). Where the mean of each marginal is 1 (i.e. $E(X_i)=1$ for all $i$). What are the restrictions on the covariance matrix (assuming that […]

A continuous mapping with the unbounded image of the unit ball in an infinite-dimensional Banach space

This question already has an answer here: Nonlinear function continuous but not bounded 3 answers

Convergence on Norm vector space.

I am not sure if this question make sense mathematically, so please bear with my ignorance. This is an extension to the question in the link: Is complete metric space is required? It seems in many engineering problem when we look for an optimal solution we work on normed vector space and it is complete […]

Measurability of supremum over measurable set

Consider a finite-valued function $f : \mathbb{R}^n \rightarrow \mathbb{R}$; and a closed-valued, measurable, set-valued mapping $S: \mathbb{R}^m \rightrightarrows \mathbb{R}^n$ . Measurability is intended with respect to a finite measure $m: \mathcal{B}(\mathbb{R}^m) \rightarrow [0,1]$, where $\mathcal{B}(\mathbb{R}^m)$ are the Borel sets. I am wondering if the following mapping is measurable as well. $$ x \mapsto \sup_{y \in […]