Functional analysis is a branch of mathematical analysis concerned with vector spaces endowed with structures that define continuity and convergence, and with mappings between such spaces. Its central objects include spaces of functions, sequences, and continuous linear operators. It combines methods from linear algebra and topology, extending finite-dimensional ideas to settings where infinitely many coordinates may be needed. The resulting framework makes it possible to study equations and approximation through properties of entire spaces rather than individual functions. (ocw.mit.edu)
Spaces and completeness
A normed vector space has a norm (|x|), which measures the size of a vector and induces the distance (d(x,y)=|x-y|). A Banach space is a normed space that is complete: every Cauchy sequence converges to an element of the space. Completeness ensures that successive approximations satisfying the Cauchy condition have a limit within the chosen mathematical setting. Scalars are usually real or complex numbers. (ocw.mit.edu)
A Hilbert space is complete under the norm induced by an inner product, [ |x|=\sqrt{\langle x,x\rangle}. ] Inner products introduce orthogonality and permit geometric constructions such as projection onto closed linear subspaces. Every Hilbert space is a Banach space, but a Banach-space norm need not arise from an inner product. (ocw.mit.edu)
Important examples include sequence spaces (\ell^p), spaces (C(K)) of continuous functions on a compact space with the supremum norm, and \(L^p\) spaces of measurable functions. For (1\leq p<\infty), [ |f|_p=\left(\int |f|^p,d\mu\right)^{1/p}. ] Elements of (L^p) are equivalence classes of functions agreeing almost everywhere; this identification makes the expression a genuine norm. The space (L^2) is Hilbert, with an inner product defined by integration. Different norms express different notions of approximation, so choosing the space is part of formulating a problem. (live.ocw.mit.edu)
Operators, functionals, and duality
A linear operator (T:X\to Y) is a linear map between vector spaces. For normed spaces, continuity is equivalent to boundedness: there is a constant (C) such that [ |Tx|_Y\leq C|x|X ] for every (x\in X). Its operator norm is [ |T|=\sup{|x|_X\leq1}|Tx|_Y. ] If (Y) is Banach, the bounded linear operators from (X) to (Y) themselves form a Banach space under this norm. (ocw.mit.edu)
A linear functional takes vectors to scalars. The continuous dual space (X^*) consists of all bounded linear functionals on (X). This differs from the algebraic dual, which includes discontinuous functionals. In a Hilbert space, the Riesz representation theorem states that every bounded linear functional is represented by taking an inner product with a unique vector. Duality thus connects scalar measurements with the geometry of a space. (ocw.mit.edu)
Fundamental theorems
Several general theorems organize linear functional analysis:
- The Hahn–Banach theorem extends a bounded linear functional from a linear subspace to the whole normed space without increasing its norm.
- The uniform boundedness principle states that a pointwise bounded family of bounded linear operators on a Banach space has uniformly bounded operator norms.
- The open mapping theorem states that a bounded, surjective linear operator between Banach spaces maps open sets to open sets. Consequently, a bounded bijective linear operator between Banach spaces has a bounded inverse.
- The closed graph theorem states that an everywhere-defined linear operator between Banach spaces is bounded if its graph is closed in their product space. (live.ocw.mit.edu)
These results turn structural assumptions—particularly completeness—into conclusions about extension, continuity, and invertibility. (live.ocw.mit.edu)
Weak convergence and compactness
Infinite-dimensional spaces differ sharply from finite-dimensional ones. Their closed unit balls are not compact in the norm topology. The weak topology provides another notion of convergence: (x_n) converges weakly to (x) when [ f(x_n)\longrightarrow f(x) \qquad\text{for every }f\in X^*. ] Norm convergence implies weak convergence, but the converse generally fails. For example, an orthonormal sequence in a Hilbert space converges weakly to zero while every vector retains norm one. (math.mit.edu)
The weak-* topology on (X^) is defined by pointwise evaluation on vectors of (X). The Banach–Alaoglu theorem states that the closed unit ball of (X^) is weak-* compact. This restores a form of compactness unavailable in the norm topology, though compactness alone does not always imply sequential compactness. (math.mit.edu)
Spectral theory
A compact operator maps bounded sets to sets with compact closure. Such operators often retain features of finite-dimensional matrices. The spectrum of a bounded operator consists of scalars (\lambda) for which (T-\lambda I) has no bounded, everywhere-defined inverse. Unlike the finite-dimensional case, spectral values need not be eigenvalues. (ocw.mit.edu)
For a compact self-adjoint operator on a Hilbert space, the spectral theorem supplies an orthonormal basis of eigenvectors; nonzero eigenvalues have finite multiplicity and can accumulate only at zero. General self-adjoint operators require a broader formulation using spectral measures. (ocw.mit.edu)
Applications
In partial differential equations, Sobolev spaces incorporate integrability conditions on functions and their weak derivatives. Weak formulations allow equations to be treated even when classical derivatives do not exist. The Lax–Milgram theorem establishes existence and uniqueness for variational problems involving bounded, coercive bilinear forms on Hilbert spaces. (dcn.nat.fau.eu)
In quantum mechanics, states are described using complex Hilbert spaces and observables by self-adjoint operators. Many important operators are unbounded, making their domains an essential part of their definition. Spectral theory provides the mathematical framework for analyzing these observables and Schrödinger operators. (asc.physik.lmu.de)