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So far, We have seen the applications of functional analysis in PDE, probability and many areas in applied mathematics. On the other hand, methods of algebraic topology are introduced to functional analysis via operator K-theory.

From the mathematical research point of view, (as a crazy fan of pure math) I wonder if functional analysis(theory of Banach and Hilbert spaces and operator algebras), as a well-built theory in mathematics, has any applications to the areas in pure math like algebraic topology, algebraic geometry and number theory, solving problems for which purely algebraic or geometric methods seem to be powerless.

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closed as too broad by Daniel Moskovich, Alexey Ustinov, Andreas Thom, Ian Morris, Yemon Choi Apr 6 '16 at 10:09

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Noncommutative geometry and the whole field-of-one-element business may be one application to number theory. $\endgroup$ – user31415 Apr 6 '16 at 5:35
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    $\begingroup$ A traditional route to applications is via Hodge theory, which has various applications in complex algebraic geometry. There is also the whole story around the index theorem and Dirac operators. $\endgroup$ – Qiaochu Yuan Apr 6 '16 at 5:51
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    $\begingroup$ My answer to this question gives six major applications of operator algebras to other areas. Of course functional analysis in general would only broaden the possible applications. $\endgroup$ – Nik Weaver Apr 6 '16 at 6:00
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    $\begingroup$ @IanMorris, some people don't have access to mathscinet. $\endgroup$ – Sergei Akbarov Apr 6 '16 at 9:53
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    $\begingroup$ To @Yemon and other users: Could you please tell me the difference between my post and the following posts?I can't see why mine is voted to close while those ones are not. I have limited my applications to pure math like algebra, geometry and number theory mathoverflow.net/questions/200696/… $\endgroup$ – No One Apr 6 '16 at 18:22
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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 typical key result in number theory whose proof relies (not only, but critically) on functional analysis is the Selberg trace formula and its variants. More generally, the theory of automorphic forms heavily depends on spectral theory, as one can see in this post in Terence Tao's blog.

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In number theory there are $p$-adic Banach spaces and $p$-adic Banach algebras (e.g., the Tate algebras), and more generally there is the whole subject of $p$-adic functional analysis. Applications of $p$-adic functional analysis go back to the 1960s in work of Dwork (as explained systematically by Serre, go here for an English translation by Jay Pottharst) on the Weil conjectures. For more recent work see here, here, and here. There is a book on this subject by Schneider.

While the Hahn-Banach theorem for real Banach spaces carries over to Banach spaces over $\mathbf Q_p$, examples of duality in the $p$-adic setting look surprising from the viewpoint of classical analysis: while the dual of $c_0(\mathbf R)$ is $\ell^1(\mathbf R)$, the dual of $c_0(\mathbf Q_p)$ is $\ell^\infty(\mathbf Q_p)$. This is due to the different nature of convergence of series in $\mathbf R$ and $\mathbf Q_p$: in $\mathbf R$ the pairing $\langle \{a_n\},\{b_n\}\rangle = \sum_{n \geq 1} a_nb_n$ if $a_n \rightarrow 0$ makes sense if $\sum |b_n| < \infty$ but not generally if $\{b_n\}$ is merely bounded (e.g., $a_n = 1/n$ and $b_n = 1$), while in $\mathbf Q_p$ the same pairing for $a_n \rightarrow 0$ does make sense if $\{b_n\}$ is bounded (for $b_n = 1$ this is the fact that a series of $p$-adic numbers converges if the general term tends to $0$).

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While I agree the question may be a little too broad as it stands, there is one rather easy answer: Hormander's $L^2$ estimate (existence theorem) for the $\bar{\partial}$-equation is powered by functional analysis (you did mention PDE), and it is one of few really general and fundamental tools in effective algebraic geometry. Major consequences in commutative algebra and in algebraic and complex geometry of the $L^2$ existence theorem include Skoda's theorem, effective Nullstellensatz and effective Artin-Rees, the analytic theory of multiplier ideals (Kohn, Nadel), the holomorphic extension theorem (Ohsawa-Takogoshi), the deformation invariance of plurigenera and finite generation of canonical rings of algebraic varieties (Siu). This paper of Siu contains a description and summary of these and other results, as of 2004.

How about number theory? Well, it is common knowledge that many of the results and problem of effective algebra and complex algebraic geometry, like the first five theorems I listed above, are eminently relevant to number theoretic questions - to those, at least, which can be expressed in the framework of Arakelov theory. Effective Nullstellensatz is needed whenever one has to quantify the bounds in Weil's height machine in order to solve a specific diophantine equation, for example. On a more sophisticated level, the Ohsawa-Takegoshi extension theorem (in a geometric form obtained by Manivel) is a crucial analytic component of the arithmetic Hilbert-Samuel formula of Arakelov theory, leading in turn to a construction of non-zero global sections of small norm for positive line bundles on arithmetic varieties. Applications include a solution of Bogomolov's conjecture for subvarieties of abelian varieties (Szpiro, Ullmo, Zhang) and the Galois equidistribution of small points in algebraic dynamics (SUZ, Yuan).

On a conceptual level I find it amusing that the construction of arithmetically effective sections needed to prove these characteristically number theoretic results (one of the jewels and triumphs of Arakelov's point of view!) are traced, via the $L^2$ existence theorem, to the basic existence principle of functional analysis: that of representing a bounded linear functional on a Hilbert space as the inner product with a fixed vector. In the $\bar{\partial}$-problem we have to solve an equation $Tf = g$ for a closed densely defined linear operator $T : H \to H'$ of Hilbert spaces (and a given $g \in H'$), with an estimate $|f| \leq C |g|$ (this being the point in effectivity). The Riesz representation theorem reduces this to proving an estimate $|\langle u,g \rangle| \leq C \cdot |T^* u|$ on all of $\mathrm{im}{\, T^*}$ (not just on a dense subspace!), and we are led in this way to functional analysis and a study of domains of adjoint operators. In its most basic form, the $L^2$ existence theorem is a simple application of this principle with the Bochner technique (the Bochner-Kodaira-Nakano identity from the proof of Kodaira's vanishing theorem).

That the $\bar{\partial}$-problem is pertinent to the holomorphic extension problem is clear once one notes that, to extend $g$ from a complex submanifold $Y \subset X$, one may start with any $C^{\infty}$ extension $h$ and then seek to solve the equation $\bar{\partial} f = \bar{\partial} h$ with the vanishing condition $f|_Y = 0$. That the extension problem is pertinent to arithmetic Hilbert-Samuel becomes obvious as soon as one contemplates the standard proof of the algebro geometric prototype. Finally, existence of the desired arithmetically effective section follows from arithmetic Hilbert-Samuel via Minkowski's theorem on lattice points in convex bodies. This way, as an arithmetic reflection of the Riesz representation principle in functional analysis, we have an arithmetic existence theorem that may be seen as a sophisticated form of the uniquitous Siegel lemma. It underlies the mentioned arithmetic applications to equidistribution, and to Bogomolov's problem: no solution is known of Bogomolov's conjecture that does not in some way touch upon these ideas.

This is not fundamentally different from the role of functional analysis in Hodge theory (mentioned in the comments), but it represents an application at once to all three domains mentioned in the question title: algebra, geometry, and number theory. So I thought I would write an answer outlining it.

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Recently a new application of functional analysis in geometry appeared, the study of envelopes of topological algebras. It allows to look at "big" geometric disciplines -- complex geometry, differential geometry, topology -- from the point of view of category theory, so that these disciplines become "purely categorical constructions". This can be considered as a developement of Klein's Erlangen program.

According to this view, different geometric disciplines are just pictures that appear in the imagination of an outlooker after applying different "observation tools" for studying a given category of topological algebras. Formally, this "projection of functional analysis to geometry" is established by a categorical construction, called envelope (and there are many different envelopes that give different geometries as disciplines).

This activity allows to build different generalizations of Pontryagin duality to classes of non-commutative groups (including some quantum groups).

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Hyper-real ideals were first introduced by functional analyst Edwin Hewitt in 1948 as part of his program of studying spaces of continuous functions. The related hyperreal fields can be taken as a basis for Abraham Robinson's framework with infinitesimals, with numerous applications in other fields, some of them mentioned here.

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