Let $G$ be a compact group. We know that the algebra of trigonometric functions is dense in $C(G)$ and hence, it separates the points of $G$. Let $X(G)$ be the linear span of all g …

Dear mathoverflowers, I have a question concerning the strong convergence in $L^p([0,T],X)$. Let $X_1,X$ be two Banach spaces such that $X_1\subset X$ with compact embedding. Let …

I am learning about Sobolev spaces on hypersurfaces. Let $S$ be a $C^k$-hypersurface with boundary for some $k$. In order to define a weak derivative, one needs $k \geq 2$ becaus …

How is the proof that $$[L^2(0,T;X)]' = L^2(0,T;X')$$ looking like, where $X$ is a Hilbert space? I am asking for the proof that the dual space of $L^2(0,T;X)$ is the space $L^2(0 …

Is it true that the space $C_c^{\infty}([0,T];V)$ is dense in $C_c^{1}([0,T];V)$? These are compactly supported functions that are $V$ valued, where $V$ is a Banach or Hilbert spac …

I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO. Let $(\Omega,\mathcal F)$ be a measurable space and $\ …

I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable …

Hello, I am looking for a paper or a book that says about the existence and some estimates (like these in the non-degenerate case) of the fundamental solutions for degenerate elli …

Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation $$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \c …

I would want to known some counterexamples for the Open Mapping Theorem, evading every single condition. I mean, we need X,Y Banach Spaces, and a linear, continous and surjective a …

I am having a problem with a certain inequality I try to understand. I think it's just a basic idia (/trick) I'm missing, but I can't seem to find it. Here's a simplification of …

An bouneded linear operator $A \in L(X, Y)$ ($X$, $Y$ Banach spaces) is called absolutely $2$-summable if there exists a $C>0$ such that $$ \left( \sum_{j=1}^N \| A x_j\|_X^2 \righ …

Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$. Is $\beta^p$, $p>2$ a tw …

My question was prompted by an earlier MO by @Daniel:     Duality map in strictly convex Banach spaces I will even use his symbol $\phi$ below. Let $B$ be an arbitrary …

Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions $$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu …