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 […]

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 $u$ […]

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 – […]

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 $\|\cdot\|_1$ […]

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) […]

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 […]

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)

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 […]

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 […]

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 $$ \varphi \;\text{$c$-topologically injective}\Longleftrightarrow\varphi^*\; \text{$c^{-1}$-topologically surjective}\\ \varphi \;\text{$c$-topologically surjective}\Longrightarrow\varphi^*\; \text{$c^{-1}$-topologically injective} $$ […]

Intereting Posts

What is the value of lim$_{n\to \infty} a_n$ if $\lim_{n \to \infty}\left|a_n+ 3\left(\frac{n-2}{n}\right)^n \right|^{\frac{1}{n}}=\frac{3}{5}$?
Null space for $AA^{T}$ is the same as Null space for $A^{T}$
Imaginary-Order Derivative
projectile motion with mass, find the range
Picking a $\delta$ for a convenient $\varepsilon$?
If $a, b, c >0$ prove that $ ^7 > 7^7a^4b^4c^4 $.
Evaluating an Integral by Residue Theorem
How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
The simple modules of upper triangular matrix algebras.
Is it true that every normal countable topological space is metrizable?
Greatest common divisor of $3$ numbers
How to prove $ \int_0^\infty e^{-x^2} \; dx = \frac{\sqrt{\pi}}2$ without changing into polar coordinates?
I want to study mathematics ahead of high school, but I found that I'm rusty on the elementary stuff
Examples of mathematical results discovered “late”
Limit with integral or is this function continuous?