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?

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 $\{x\in H\colon \|x\|\leqslant 1\}$ is compact, then how can it be proved that $H$ is finite-dimensional?

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 […]

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.

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” […]

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) […]

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 […]

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 ?

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. […]

Intereting Posts

Euler numbers grow $2\left(\frac{2}{ \pi }\right)^{2 n+1}$-times slower than the factorial?
Continuous images of compact sets are compact
Closed form of $\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$
fundamental group of the complement of a circle
Is there much of difference between set models and class models?
How does FFT work?
Is the exclusion of uncountable additivity a drawback of Lebesgue measure?
Can $\mathbb R$ be written as the disjoint union of (uncountably many) closed intervals?
A set with a finite integral of measure zero?
number of subsets of even and odd
Intuition behind Matrix Multiplication
Inverse function of $x^x$
How to prove a trigonometric identity
how to show that a group is elementarily equivalent to the additive group of integers
Evaluate the limit: $\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$