TITLE: Using Functional Analysis for Differential Equations
QUESTION [7 upvotes]: In the functional analysis books that I have read, they do not explain how the ideas and theorems of functional analysis (in the sense of operators on Banach spaces) help to deal with differential equations, such as proving existence or uniqueness of solutions. 
Could someone give me an example of using the ideas and theorems of functional analysis to actually say something about a (partial) differential equation?

REPLY [4 votes]: There are in fact many applications of functional analysis and functional analytic techniques to the theory of differential equations, both ordinary and partial.  Indeed, so many such applications exist that it is impossible to even begin to provide a comprehensive list in such a small space.  Therefore I will focus on two of most central, one in each of the fields ordinary and partial differential equations.
To begin with an example of how functional analytic techniques come to play in the theory of ordinary differential equations, we may turn to the Picard-Lindeloef theorem, which is the central result affirming the existence and uniqueness of solutions to a large and important class of ODEs, namely those of the form
$\dot{\vec y} = f(\vec y, t), \tag 1$
where $f(\vec y, t)$ is assumed to be Lipschitz continuous in $\vec y$ and jointly continuous in $\vec y$ and $t$.  Picard-Lindeloef is typically proved via an invocation of the Banach fixed point theorem, also sometimes known as the contraction mapping theorem, which I think may be properly regarded as within the domain of functional analysis per se, although some may disagree.  Allow me to explain:
A major program of functional analysis is to study objects which are infinite-dimensional generalizations of their finite dimensional counterparts; thus we work with   normed, Banach and Hilbert spaces etc., and general linear operators between them in lieu of more specialized functions such as derivatives and integrals and so forth.  Of course, we still draw upon these operations from ordinary calculus, or even real and complex analysis, in order to map out just where a fruitful application of a more functional analytic approach might be found.  It strikes me that a fruitful basic example here is the isometric isomorphism between $L^2(\Bbb R)$ and itself provided by the Fourier transform; here we see both a very specific integral operator pf the form
$\mathcal F(f)(\omega) = \displaystyle \int_{-\infty}^\infty f(x) e^{-i\omega}x \; dx, \; f(x) \in L^2(\Bbb R) \tag 2$
as well as an isometric linear map.  The Fourier transform in some sense lives at the boundary of real and complex analysis and the theory of operators on Hilbert spaces; many aspects of functional analysis dwell at similar boundaries between otherwise seemingly disparate subjects; indeed, it is the functional analytic approach which unifies such differing approaches to certain problems.
So how does the Banach fixed point theorem fit into the general program of functional analysis?  Well, for one thing, it pertains to functions 
$T:X \to X, \tag 3$
where $X$ is a complete meric space, such that there exists some $k$,
$0 < k < 1, \tag 4$
such that
$x_1, x_2 \in X \Longrightarrow d(f(x_1)), f(x_2)) < k d(x_1, x_2), \tag 5$
where
$d:X \times X \to \Bbb R \tag 6$
is the metric on $X$; the theorem then affirms the existence of a unique point
$x^\ast \in X \tag 7$
such that
$\displaystyle \lim_{i \to \infty} f^i(x) = x^\ast \tag 8$
for any $x \in X$.  We note that many linear spaces provide us with subspaces such as the complete metric space $X$ (note that we do not stipulate $X$ itself be a linear subspace); therefore it should come as no surprise that the usual linear spaces encountered in analysis often contain (not necessarily linear) subpaces $X$ and maps $T:X \to X$ to which the fixed point theorem applies.  
In fact, perhaps the premiere application of the Banach fixed point theorem in the field of ordinary differential equations is to the proof of the above-mention Picard-Lindeloef theorem; in this case, assuming
$\dim \vec y = n, \tag 8$
and a solution to (1) is sought on some interval
$[a, b] \subset \Bbb R, \tag 9$
with the specified initial condition
$\vec y(a) = \vec y_0, \tag{10}$,
we look at the set
$X = \{ \vec y(t) \in C^0([a, b], \Bbb R^n, \; \vec y(a) = \vec y_0 \}, \tag{11}$
and define
$T: X \to X \tag{12}$
via
$T(\vec y(t)) = \vec y_0 + \displaystyle \int_a^t f(\vec y(s), s) \; ds; \tag{13}$
for
$\vec y_1(t), \vec y_2(t) \in X; \tag{14}$
we have, assuming $k$ is a Lipschitz constant for $f$, that is
$\Vert f(\vec y_1, t) - f(\vec y_2, t) \Vert \le k \Vert y_1 - y_2 \Vert, \tag{14.5}$
$\Vert T(y_2(t)) - T(y_1(t)) \Vert = \displaystyle \sup_{t \in [a, b]} \vert T(y_2(t)) - T(y_1(t)) \vert$
$= \displaystyle \sup_{t \in [a, b]} \left \vert \displaystyle \int_a^t (f(\vec y_2(s), s) - f(\vec y_1(s), s)) \; ds \right \vert \le  \displaystyle \sup_{t \in [a, b]} \int_a^t \vert f(\vec y_2(s), s) - f(\vec y_1(s), s) \vert \; ds$
$= \displaystyle \int_a^b \vert f(\vec y_2(s), s) - f(\vec y_1(s), s) \vert \; ds   \le \displaystyle \int_a^b k \vert y_2(s) - y_1(s) \vert \; ds$
$\le k \displaystyle \int_a^b \sup_{t \in [a, b]} \vert \vec y_2(t) - \vec y_1(t) \vert \; ds = k \displaystyle \int_a^b \Vert \vec y_2(t) - \vec y_1(t) \Vert \; ds = k(b - a)\Vert \vec y_2(t) - \vec y_1(t) \Vert \; \tag{15}$
this shows that $T$ is a contraction mapping if 
$k(b - a) < 1, \tag{16}$
and hence under this condition (13) has a unique fixed point such that
$\vec y(t) = \vec y_0 + \displaystyle \int_a^t f(\vec y(s), s) \; ds; \tag{17}$
upon differentiating this equation we obtain
$\dot{\vec y(t)} = f(\vec y(t), t), \tag{18}$
showing such $\vec y(t)$ is the unique solution to (1).
Thus we see how functional analytic techniques come to play in the theory of ordinary differential equations.
Applications to partial differential equations are perhaps even more obvious.  I think here, for example, of the Babuška–Lax–Milgram_theorem and its relation to weak solutions of certain PDEs, as is outlined in the answer of pseudocydonia.  Since this answer is already perhaps overlong, I leave it to the reader to invesitgate the link I have provided to Babuška–Lax–Milgram for a more complete picture of this application of functional analysis to PDEs.