Articles of functional analysis

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. […]

Is the derivate on a closed subspace of $C^1$ is a continuous linear map?

I’m trying to show that $D:(X, \|\cdot\|_\infty) \rightarrow C[0,1]$ is a continuous map. $D$ is the differential operator and $X$ is a closed (proper) subset of $C^1[0,1]$. The fact that $X$ is closed in $C^1[0,1]$ must be important in the proof because otherwise this result is obviously false. However, I don’t know how to use […]

Equicontinuity if the sequence of derivatives is uniformly bounded.

I would really appreciate if someone could look over this proof for me. Let $ \left\{ g_m \right\} $ be a sequence of functions defined on an interval $ [a,b] \subset \mathbb{R}^n$. Let $ \left\{ g’_m \right\} $ be uniformly bounded on $[a,b]$. Show that $ \left\{ g_m \right\} $ is equicontinuous on $[a,b]$. My […]

If $Q$ is an operator on a Hilbert space with $Qe_n=λ_ne_n$ for all $n$, then $Q^{-\frac 12}e_n=\frac 1{\sqrt{λ_n}}e_n$ for all $n$ with $λ_n>0$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $\mathfrak L(U)$ be the set of bounded and linear operators on $U$ $Q\in\mathfrak L(U)$ be nonnegative and symmetric $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for all }n\in\mathbb N\tag 1$$ for some $(\lambda_n)_{n\ge 0}\subseteq[0,\infty)$ We can prove, that for any nonnegative and symmetric $L\in\mathfrak L(U)$ there […]

Transpose of Volterra operator

I want to find the transpose of the Volterra operator $$Vf(x) = \int_0^x f(t)dt, \;\; x\in(0,1)$$ acting in $V:L^2(0,1) \rightarrow V:L^2(0,1) $. The transpose is defined as $\textbf{M}’:U’\rightarrow V’$. For Hilbert spaces the transpose is replaced by the adjoint. I would guess that the transpose is also a map $\textbf{M}’:L^2(0,1)\rightarrow L^2(0,1)$ Since $L^2$ is self […]

How to prove Campanato space is a Banach space

Let $p\ge1$, $\mu\ge0$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$. Campanato space embraces all $u$’s which $$[u]_{p,\mu}=[u]_{p,\mu;\Omega}=\sup_{\substack{x\in\Omega\\0<\rho<\mathrm{diam}\Omega}}\left(\rho^{-\mu}\int_{\Omega_\rho(x)}\left|u(y)-u_{x,\rho}\right|^p\,dy\right)^{\frac{1}{p}}<+\infty,$$ where $\Omega_\rho(x)=\Omega\cap B_\rho(x)$ ($B_\rho(x)$ denotes a ball centered at $x$ with a radium $\rho$) and $$u_{x,\rho}=\frac{1}{\left|\Omega_\rho\right|}\int_{\Omega_\rho(x)}u(y)\,dy, $$ equipped with a norm $$\|u\|_{L^{p,\mu}}=\|u\|_{L^{p,\mu}(\Omega)}=[u]_{p,\mu;\Omega}+\|u\|_{L^p(\Omega)}.$$ Let $\{u_k\}$ be a Cauchy sequence in Campanato space, one can determine a $u$ […]