Functional analysis takes the ideas of continuity, completeness and convergence that were developed for numbers and functions of one variable and applies them wholesale to spaces of functions, operators and sequences, and it is the framework in which the modern theory of differential equations, Fourier methods and quantum mechanics is written. A normed vector space equips its elements with a notion of length and therefore of distance, and a space is called complete, or a Banach space, when every Cauchy sequence of its elements converges to an element of the space. This single condition, abstract as it looks, is the setting in which convergence questions become decidable and in which existence theorems can be proved without any appeal to concrete formulas. The spaces that matter most are familiar under other names: the continuous functions on a compact interval with the uniform norm, the integrable functions with the norm given by the integral of the absolute value, the finite dimensional spaces of any dimension, and the Hilbert spaces whose norm comes from an inner product, of which the space of square-integrable functions is the prototype and the one that most directly generalises ordinary Euclidean geometry. A linear map between such spaces is continuous precisely when it is bounded, that is, when the size of its output is at most a fixed multiple of the size of its input, and the set of bounded operators is itself a Banach space under the operator norm, so that the algebra of operators becomes a subject in its own right. The structural theorems of the subject are few in number and immense in reach. The contraction mapping principle says that a map from a complete metric space into itself which shrinks all distances by a factor less than one has a unique fixed point, and its proof is a one-line geometric series argument, yet it alone yields the fundamental existence and uniqueness theorem for ordinary differential equations, the convergence of the Picard iteration, and the contraction argument behind the Newton-Kantorovich theorem for solving nonlinear algebraic systems. Boundedness principles, in the form of the Banach-Steinhaus theorem and the open mapping theorem, then convert pointwise information into uniform control, and the Hahn-Banach theorem extends bounded linear functionals from subspaces to the whole space, which is the analytic source of the separation arguments used in optimisation and convexity. Spectral theory occupies the centre of the subject. On a finite dimensional space an operator has finitely many eigenvalues, but the Hilbert space analogue has a possibly continuous spectrum, and the theorem for compact self-adjoint operators asserts that such an operator is described by an orthonormal sequence of eigenfunctions with real eigenvalues, a statement that looks deceptively like the elementary diagonalisation of a symmetric matrix and is behind spectral resolution, the harmonic analysis of the Fourier transform, and the classification of normal modes in quantum mechanics. Unbounded operators must be handled with care, since differentiation is not continuous in the uniform norm, and the modern resolutions are the theory of closed operators and of distributions, in which derivatives act on test functions rather than on functions. Applied mathematics meets functional analysis most directly in the calculus of variations and the theory of weak solutions. Many physical problems, among them the equilibrium of an elastic membrane and the stationary states of quantum mechanics, are posed by asking for a function that minimises an energy functional, and the Euler-Lagrange equation is the necessary condition obtained by varying the function slightly; the second variation then decides stability. When the admissible functions are not smooth, the minimiser is only determined up to values on a set of measure zero, and this is where distributions and Sobolev spaces enter, admitting functions whose derivatives exist in the weak sense. Lax and Milgram, a compactness argument built on the Hilbert space framework, guarantees existence and uniqueness for a large class of linear boundary value problems, and the finite element method is the numerical shadow of the whole structure: choose a function space of piecewise polynomial trial functions, restrict the energy to a finite dimensional subspace, assemble a stiffness matrix from the weak formulation, and solve the resulting linear system. Anyone who has implemented a finite element code, or who has read a proof of well-posedness for a partial differential equation, has used functional analysis without necessarily naming it, which is the clearest evidence of how thoroughly the subject has penetrated applied work.
