Articles of functional analysis

Projections: Ordering

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P’^2=P’=P’^*$$ Order them by: $$P\leq P’:\iff\sigma(\Delta P)\geq0\quad(\Delta P:=P’-P)$$ Then equivalently: $$P\leq P’\iff P=PP’=P’P\iff\Delta P^2=\Delta P=\Delta P^*$$ How can I check this? (Operator algebraic proof?)

Basis for a product Hilbert space.

Let $H_1, H_2, \ldots, H_n$ be a countable family of Hilbert spaces. Let H be the set of tuples $x = (x_1, \ldots, x_n,\ldots)\in \prod_n H_n$ with the property that $$\|x \| ^2 =\sum_n \| x_n \| _{H_n}^2 <\infty.$$ Then H is also a Hilbert space. Prove that H is non-separable and determine an orthonormal […]

What do we need Sobolev-spaces $W^{k, p}$ with $p \neq 2$ for?

In the course on PDE’s I took this semester we talked a lot about the theory of Sobolev-spaces $W^{k, p}(\Omega)$ for $\Omega \subset \mathbf{R}^n$ an open set, $k \in \mathbf{N}$ and $1 \leq p \leq \infty$. But then we only used the Sobolev-spaces with $p = 2$ to deal with PDE’s, since we can use […]

Nontrivial solutions of $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$

Let $a=(a_n)$ with $a_n\in\mathbb{C}$ be a vector indexed over all $n\in\mathbb{Z}$, and consider the system of equations $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$ for all $k\in\mathbb{Z}$. One may verify this has a family of trivial solutions given by $a_n=\delta_{nm}$ for some nonzero integer $m$. Assuming $a_0=0$, are there any other solutions? This problem is inspired by (unsuccessful) attempts to find […]

Is every irreducible operator unitary equivalent to a banded operator?

This issue continues this question. Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis. $T \in B(H)$ is banded if $\exists r \in \mathbb{N}$ such that $ (Te_{n}, e_{m})\ne 0 \Rightarrow \vert n-m \vert \leq r$. Definition : […]

$fg\in L^1$ for every $g\in L^1$ prove $f\in L^{\infty}$

Let $(X,\mathcal{A}, \mu)$ be an arbitrary measure space. Let $f$ be an extended complex-valued $\mathcal{A}-$measurable function on $X$ such that $|f|<\infty$ $\mu$-a.e. on $X$. Suppose that $fg\in L^1(X,\mathcal{A}, \mu)$ for every $g\in L^1(X,\mathcal{A}, \mu)$. Show that $f\in L^{\infty}(X,\mathcal{A}, \mu)$. Can anyone verify my answer? Does anyone know a better elementary approach? (It’ll be great if […]

The spectrum of a commutative unital Banach algbra having linearly dense idempotents is totally disconnected

I don’t know how should I start to show: If $A$ is an arbitrary abelian Banach algebra in which the idempotents have dense linear span, its specrum (the space of characters on $A$) is totally disconnected.

The compactness of the unit sphere in finite dimensional normed vector space

We define $ (\mathbb{R}^m, \|.\|)$ to be a finite dimensional normed vector space with $ \|.\|$ is defined to be any norm in $ \mathbb{R^m}$. Let $S = \lbrace x \in \mathbb{R}^m: \| x\| = 1 \rbrace.$ Prove that $S$ is compact in $ (\mathbb{R}^m, \|.\|).$

Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma

This question already has an answer here: Hahn-Banach Theorem for separable spaces without Zorn's Lemma 2 answers

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 […]