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Huge chunks of the theory of nonlinear PDEs rely critically on analysis in $L^p$-spaces. Let's take the 3D Navier-Stokes equations for example. Leray proved in 1933 existence of a weak solution to …

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 …

In many ways $H^1$ is just a natural substitute for $L^1$. A typical $H^1$ function is a $1$-atom, i.e. a function $\phi\in L^1(\mathbb R^n)$ such that the support of $\phi$ is contained in some bal …

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\ …

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 $J$ …

Compensated compactness helps when one needs to find the limit of $u_n \cdot v_n$, where the sequences of vector fields $u_n$ and $v_n$ converge weakly in $L^2$: $u_n\rightharpoonup u$, $v_n\righth …

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 …

According to the Schwartz Kernel Theorem and its variants, there are the canonical isomorphisms $$\mathcal{D}^{\prime} \left(X\right)\tilde\otimes \mathcal{D}^{\prime} \left(Y\right)\simeq\mathcal{D …

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 …

"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 …

I would say these concepts rather belong to the field of potential theory. You will find most of the definitions and a fairly advanced treatment of the subject in Logarithmic Potentials with External …

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 …

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 …