Articles of functional analysis

Convex hull of bounded set is bounded

I want to show that in a locally convex topological vector space $X$, the convex hull of a bounded set is bounded. Apparantly this does not hold if $X$ is not locally convex. So the fact that that there exists a local base of absolutely convex absorbent sets is crucial. Thanks for suggestions or tips.

Kernel of the closure of an unbounded operator

Let $T:\mathcal{D}(T)\subset X\to Y$ be a densely defined closable operator. Define $\text{ker}\,T=\{(x,0)\in \text{graph}\,T\}$ where $\text{graph}\,T\subset X\times Y$. My question is $\overline{\text{ker}\,T}=\text{ker}\,\overline{T}$? I recall that $\overline{T}$ is the closure of $T$. I know that $\overline{\text{ker}\,T}\subset\text{ker}\,\overline{T}$, what about the reverse inclusion? So far I have proved that $\text{ker}\,T^{\perp}\subset R(T^*)$ implies the inclusion $\overline{\text{ker}\,T}\supset\text{ker}\,\overline{T}$, but this inclusion seems […]

Finding the adjoint for a diagonal operator in $\ell^2$.

Let $\{e_n\}_{n=1}^{\infty}$ be an orthonormal basis of the complex Hilbert space $\ell^2$. Fix complex numbers $\lambda_1,\lambda_2,\lambda_3,\dots$, let $$ \mathscr{D}(T)=\{\sum_{n=1}^{\infty} x_n e_n\in \ell^2:\sum_{n=1}^{\infty}|\lambda_n x_n|^2< \infty\} $$ and define $T:\mathscr{D}(T) \rightarrow \ell^2$ as $$ T\left( \sum_{n=1}^{\infty} x_n e_n \right) = \sum_{n=1}^{\infty} \lambda_n x_n e_n $$ for $\sum_{n=1}^{\infty} x_n e_n \in \mathscr{D}(T)$. Determine the adjoint $T^*$ of $T$. […]

Strictly monotone probability measure

Let $m$ be a probability measure on $X \subseteq \mathbb{R}^n$. Let $f: X \rightarrow \mathbb{R}$ be measurable. Assume that there exists $\epsilon > 0$ such $m\left( \{ x \in X \mid |f'(x)| \leq \epsilon \}\right) = 0$, that is, $f$ is almost flat on a set of measure $0$. Under what conditions, for all $\alpha\geq […]

Uncountable disjoint union of measure spaces

Let $(a,b)$ be an interval. Let $(A_i, \Sigma_i, \mu_i)$ be a measure space for each $i \in (a,b)$. Is it possible to put a measure space on the disjoint union $$\bigcup_{i \in (a,b)}\{i\}\times A_i?$$ If the $A_i = \Omega_i$ were open subsets of $\mathbb{R}^n$, we can think of this disjoint union as a non cylindrical […]

Definition of Sobolev Space

I have a definition that says that the space of functions that satisfy$$\|u\|_{H^m}^2=\sum_{k\in\mathbb{Z}}(1+|k|^2)^m|\hat{u}_k|^2<\infty$$is called Sobolev Space and when $m=1$, this is equivalent to saying, $$\int_0^{2\pi}(|u(x)|^2+|u'(x)|^2)dx<\infty$$ I have a few questions. The Sobolev Space is about the space of functions whose functions have well behaved derivatives in some sense (??). Or at least the norm incorporates […]

Locally convex topological vector space

In $C(\mathbb{R})$ space we define two families of seminorms: $p_x(f)=|f(x)|$ where $ x\in\mathbb{Q}$ and $q_x(f)=|f(e^x)|$ where $ x \in \mathbb{R}$ I have to check if above families induce locally convex topological vector space in $C(\mathbb{R})$. I have also to check the continuity of functional $C(\mathbb{R}) \ni f \rightarrow f(\sqrt{2}) \in \mathbb{R}$ in any of those […]

A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$

As the title says, the question is how to prove A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for some $T\in B(X,Y)$

Can $C(\mathbb R)$ be reflexive?

As in title, I was wondering whether $C(\mathbb R)$ was reflexive (here $C(\mathbb R)$ is meant as the space of continuous functions on $\mathbb R$, without any other condition). This question is generated by the following well-known result: Proposition. $(C(K), \| \, \|_\infty)$, $K$ infinite compact metric space, is nonreflexive. This a simple consequence of […]

Is tensor product of Sobolev spaces dense?

My question is: is $W_2^k(\mathbb{R})\otimes W_2^k(\mathbb{R})$ dense in $W_2^k(\mathbb{R}^2)$, and more generally is this true in $\mathbb{R}^d$? I found this post: Tensor products of functions generate dense subspace? which shows the above type of result for $C_c^\infty$. So my guess is that the answer should be affirmative, maybe requiring the assumption that $k>d/2$?