Articles of functional analysis

Normal operator matrix norm

I have some troubles to show that the operator norm of a normal operator is always equal to its largest eigenvalue, how can I proof this? Does anybody of you have a hint? My problem is, that I do not know where to use that this operator is normal?

‎Jointly ‎continuous of product in $B(H)$

‎Let ‎$‎B(H)‎$ ‎be the set of‎ ‎bounded ‎operators ‎on a Hilbert space ‎‎$‎‎H$.‎ ‎ I ‎know that ‎$u_{‎\alpha‎}‎\longrightarrow u‎$‎‎‎ ‎in ‎S.O.T ‎if ‎and ‎only ‎if‎ ‎$u_{‎\alpha‎}(x)‎\longrightarrow u(x)‎$ for all‎$‎‎x\in H$.‎ ‎ I know that ‎$\cdot : ‎‎B(H)‎\times B(H) ‎\longrightarrow ‎B(H)‎$‎‎‎ ‎such ‎that ‎‎$‎‎(u,v)‎\longmapsto uv‎$‎‎ ‎is ‎separately ‎continuous ‎and ‎jointly ‎continuous ‎on ‎bounded ‎set. ‎ ‎ […]

In a Hilbert space $H$, if the closed unit ball is compact, then how can it be proved that $H$ is finite-dimensional?

In a Hilbert space $H$, if the closed unit ball $\{x\in H\colon \|x\|\leqslant 1\}$ is compact, then how can it be proved that $H$ is finite-dimensional?

Compact operators, injectivity and closed range

Let $X$ be a an infinite dimensional Banach space. $A\in B(X)$ is a compact operator. If its range $\operatorname{Im}(A)$ is closed in $X$ then $A$ cannot be injective because $A:X\to \operatorname{Im}(A)$ would be a compact bijection between Banach spaces and the unit ball $B_X=A^{-1}AB_X$ would be compact. Now if $A$ is not injective, can we […]

Is the countable direct sum of reflexive spaces reflexive?

Let $(X_n)$ be a sequence of reflexive spaces. We define the $\ell_2$-direct sum $\bigoplus_n X_n$ as the normed space with elements $(x_n)\in \prod_n X_n$ such that $$ \|(x_n)\|=\left(\sum_{n}\|x_n\|^2\right)^{\frac{1}{2}}<\infty. $$ Is $\bigoplus_n X_n$ reflexive? What if we define analogously a $\ell_p$-direct sum for a different $p$? Thank you.

Weak convergence of a sequence of characteristic functions

I am trying to produce a sequence of sets $A_n \subseteq [0,1] $ such that their characteristic functions $\chi_{A_n}$ converge weakly in $L^2[0,1]$ to $\frac{1}{2}\chi_{[0,1]}$. The sequence of sets $$A_n = \bigcup\limits_{k=0}^{2^{n-1} – 1} \left[ \frac{2k}{2^n}, \frac{2k+1}{2^n} \right]$$ seems like it should work to me, as their characteristic functions look like they will “average out” […]

Continuous inclusions in locally convex spaces

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C \left \| x \right \|_X$ for some $C \geq 0$ (general definition of continuous inclusions in normed spaces) […]

Rellich's theorem for Sobolev space on the torus

From John Roe: Elliptic operators, topology and asymptotic methods, page 73: Let $H^{k}$ be the Soblev space defined on the torus $\mathbb{T}^{n}$ with the discrete $k$-norm: $$ \langle f_{1}, f_{2}\rangle_{k}=(2\pi)^{k}\sum_{v\in \mathbb{Z}^{n}}\tilde{f}_{1}(v)\overline{\tilde{f}_{2}}(v)(1+|v|^{2})^{k} $$ John Roe claimed that there is a Rellich type compact embedding theorem available. If $k_{1}<k_{2}$, then the inclusion operator $H^{k_{2}}\rightarrow H^{k_{1}}$ is a […]

On continuously uniquely geodesic space II

This question was inspired by this answer of @wspin. Definition : A continuously uniquely geodesic space is a uniquely geodesic space whose geodesics vary continuously with endpoints. Question : Is a complete uniquely geodesic space, continuously uniquely geodesic ?

Density of $\mathcal{C}_c(A\times B)$ in $L^p(A, L^q(B))$

Let $A, B$ be two open sets in $\mathbb{R}^n, \mathbb{R}^m$ respectively and denote $\mathcal{C}_c(A\times B)$ the space of continuous functions with compact support in $A\times B.$ Is $\mathcal{C}_c(A\times B)$ dense in $L^p(A, L^q(B))$ for any $+\infty > q,p \geq 1 ?$ I believe that the answer is YES and I’m looking for a simple proof. […]