Articles of functional analysis

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 $u$ […]

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 3: What is the norm of this functional?

What is the norm of the linear functional $f$ defined on the normed space $C[a, b]$ of all functions defined and continuous on the closed interval $[a,b]$ with the norm defined as $$\Vert x \Vert \colon= \max_{t\in[a,b]} \vert x(t) \vert \; \; \; \forall x \in C[a,b]?$$ Let $$ f(x) \colon= \int_a^{\frac{a+b}{2}} x(t) dt – […]

How can I show that it's a Banach space?

Let $I=[a,b]$ (where $a<b$) be a compact interval on $\mathbb{R}$, $0<\alpha\leq1$. and $$\mathrm{Lip}(\alpha)=\left\{f:I \to \mathbb{C} \;\bigg|\; M_f=\sup_{s\neq t} \frac{|f(s)-f(t)|}{|s-t|^{\alpha}} < \infty \right\}$$ 1) Show that $\mathrm{Lip}(\alpha)$ space is a Banach space with the norm $\|f\|_1=\sup_{t \in I}|f(t)|+M_f$. 2) Show that $\mathrm{Lip}(\alpha)$ space is also a Banach space with the norm $\|f\|_2=|f(a)|+M_f$. Hint: Show that $\|\cdot\|_1$ […]

Lax-Milgram theorem on Evans. If the mapping is injective why do we need to prove uniqueness again?

This is the theorem and its proof (From Evans L., Partial Differential Equations, p. $297-299$) If we already know that $\langle f,v \rangle = (w,v)$ = $B[u,v] = (Au, v)$ and we know that $A$ is one-to-one, isn’t that already a proof that there can be only one $u$ for which $$ B(u,v) = (w,v) […]

How to show pre-compactness in Holder space?

Let $K \in \mathbb{R}^d$ be a compact set and consider the space of Hölder continuous functions $C^{0,\gamma}(K)$ with norm $||f||_{C^{0,\gamma}}:=||f||_{\infty}+\sup_{x,y \in K,x \neq y}\frac{|f(x)-f(y)|}{|x-y|^{\gamma}}$. Assume we have a bounded sequence $\{f_n\} \subset C^{0,\gamma}(K)$, i.e. $\exists C>0$ s.t. $\sup_{n}||f_n||_{C^{0,\gamma}} \leq C$. Under what conditions can we say the sequence $\{f_n\}$ is pre-compact in $C^{0,\gamma}(K)$? In other […]

Are $L^\infty$ bounded functions closed in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a closed subset of $L^2$? (Closed in the topology induced by the $L^2$-norm)

Relationship between $C_c^\infty(\Omega,\mathbb R^d)'$ and $H_0^1(\Omega,\mathbb R^d)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ and $$H:=\overline{\mathcal D}^{\langle\;\cdot\;,\;\cdot\;\rangle_H}\color{blue}{=H_0^1(\Omega,\mathbb R^d)}$$ with $$\langle\phi,\psi\rangle_H:=\langle\phi,\psi\rangle+\sum_{i=1}^d\langle\nabla\phi_i,\nabla\psi_i\rangle\;\;\;\text{for }\phi,\psi\in\mathcal D$$ How are the topological dual spaces $\mathcal D’$ and $H’$ of $\mathcal D$ and $H$ related? Let me share my thoughts and please correct me, if I’m wrong somewhere […]

Normed vector space with a closed subspace

Suppose that $X$ is a normed vector space and that $M$ is a closed subspace of $X$ with $M\neq X$. Show that there is an $x\in X$ with $x\neq 0$ and $$\inf_{y\in M}\lVert x – y\rVert \geq \frac{1}{2}\lVert x \rVert$$ I am not exactly sure how to prove this. I believe since $M\neq X$ we […]

Proof of implication: $\varphi^*\text{ is bounded below}\implies\varphi\text{ is a quotient map}$

We say that a bounded operator $\varphi:X\to Y$ is $c$-topologically injective if $\Vert\varphi(x)\Vert\geq c\Vert x\Vert$ for all $x\in X$ $c$-topologically surjective if for all $y\in Y$ there exist $x\in X$ such that $\varphi(x)=y$ and $\Vert x\Vert\leq c\Vert y\Vert$ I have already proved that $$ \varphi \;\text{$c$-topologically injective}\Longleftrightarrow\varphi^*\; \text{$c^{-1}$-topologically surjective}\\ \varphi \;\text{$c$-topologically surjective}\Longrightarrow\varphi^*\; \text{$c^{-1}$-topologically injective} $$ […]