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Besides Hodge and index theories, mentioned in Qiaochu Yuan's comment above as applications of functional analysis to (complex) algebraic geometry and algebraic topology respectively, I believe that a …

This seems to be one special case of the so called Fourier restriction problem (see, for instance, Terry Tao's discussion: http://terrytao.wordpress.com/2010/12/28/the-bourgain-guth-argument-for-provi …

Let $\Omega\subset\mathbb{R}^n$ be open, $\mathscr{C}(\Omega,\mathbb{R})$ the Fréchet space of real-valued continuous functions on $\Omega$ endowed with the compact-open topology, and $\mathscr{C}_u(\ …

This is not really an answer but rather a long-ish comment. First of all, if $M$ is not compact, $\mathfrak{A}=C(M,\mathbb{R})$ is not really a C${}^*\!$-algebra but actually only a locally C${}^*\!$- …

I'll try to explain what Igor meant in his comments in a different way, maybe it helps. Of course, any tempered distribution is a distribution in the broader sense - more precisely, any compactly sup …

Since $\mathscr{S}'$ is the dual of an infinite-dimensional Fréchet space, the weak-* topology in $\mathscr{S}'$ is not even first countable. What people usually do is to define probability measures i …

I will leave to Yemon Choi discussing the answer from Gelfand-Raikov-Shilov's book (Commutative Normed Rings, I suppose?), and restrict myself to more recent discussions on the matter... There is an …

Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given …

I doubt it. My reason is related to your comment: take your favorite function $f$ on $\mathscr{D}$, multiply it by a Gaussian with covariance $\sigma$ and centered around a point $x$ in the support of …

The "infinite tensor product" picture may be useful as a sort of concrete image of the state space of a quantum field theory, but in practice is rarely used because of the technical difficulties it br …

Let $0\leq f\in\mathscr{D}(\mathbb{R}^n)$. As shown e.g. by J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006) 137-14 …

Let $K\subset\mathbb{R}^d$ be a compact set with non-empty interior and Lipschitz boundary. In Section VI.3 of his book "Singular Integrals and Differentiability Properties of Functions", E. M. Stein …

Let $X$ be a Hausdorff locally convex vector space. Recall (my reference is the book of H. Jarchow, Locally Convex Spaces. B.G. Teubner, 1981) that we say that $X$ is a semi-Montel space if every boun …

This is easy and comes from the fact that $V$, being a Fréchet space, is barrelled, that is, the locally convex topology of $V$ coincides with its strong topology $\beta(V,V')$ (where $V'$ = topologic …

Not always. What you are talking about is also called "convergence in the sense of sesquilinear forms", because you are taking a pointwise limit in $D\times D$ of a sequence of sesquilinear forms $\al …