Articles of functional analysis

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 $u$ […]

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 […]