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The argument you have presented is an adaptation of the Lax-Milgram theorem which is essentially equivalent to the Riesz representation theorem (and generally speaking both of these results hold only …

For any locally convex and metrizable space $E$, its strong dual is metrizable if and only if $E$ is normable. This and related properties of (F)-spaces are discussed in detail in Topological Ve …

Littlewood's $4/3$-inequality singles out $\ell^{4/3}$. Namely, given a real valued array $\hat{a}=(\hat a_{m,n}:(m,n)\in\mathbb N^2)$, the norm $\|\hat a\|_{\ell_p}$ is finite for all $\hat a$ suc …

Let me try to answer the question how I understand it (basically, just to expand a bit on the comments by Harald and Willie). Let $Df$ be a distributional derivative of a differentiable function $f: …

"Topological Vector Spaces" by Helmut Schaefer contains a thorough treatment of the classical Krein-Rutman theorem for compact positive operators in an ordered Banach space along with several general …

I'd like to expand a bit on Pietro Majer's remark concerning the relation with the principle of uniform boundedness. Indeed, suppose that the trigonometric system is a Schauder basis of $C(\mathbb T …

Question. Is there a concrete example of a bounded linear operator on a Hilbert space for which it is not known if it has a non-trivial closed invariant subspace? [Added 24.01.2011: According to Bern …

The problem is nontrivial already in the finite dimensional case $E= \mathbb R^d$, $F=\mathbb R$. The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\ …

Let X be a metric space. Then every Borel measure μ on X is regular (i.e. for every Borel set B and every ε > 0, there exists a closed set $F_ε$ such that $F_ε ⊂ B$ and μ(B\ $F_ε$) < ε). If X is comp …

This is a huge subject. The minimum sizes of $\epsilon$-nets of compacts in linear spaces were studied by Kolmogorov and his school. They showed that in general there are no good bounds for this quan …

This is true. By a partition of unity, the proof can be reduced to the case when the test functions have their supports in a unit cube and the result follows from a more or or less straightforward …

If you are interested in $L^p$-theory, you are probably looking for solutions belonging to a Sobolev class $H^{s,p}(\Omega)$ with some $s>0$ and $p>1$. In this case, the Besov space $B^{s-1-1/p,p}(\p …

An operator-theoretic proof that the classical Hamburger moment problem admits a solution (see e.g. Methods of modern mathematical physics by Reed and Simon, volume 2, Theorem X.4). Weyl's proof of …

A counterexample for both the first and third questions can be found in Counterexamples in Topological Vector Spaces by Khaleelulla (p. 108). Let $W$ denote the space of all $\mathbb C$-valued seque …

The classical electric current density can be modelled as a 2-form $$J=J_{ij}\wedge dx^{ij}$$ which is assumed to be locally integrable over a 3-manifold (3-dimensional domain) $X$. By integrating $J$ …