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. […]

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 […]

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 […]

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 […]

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 […]

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