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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