About This Course

Functional analysis is linear algebra carried out in spaces whose vectors are sequences and functions. In such spaces dimension is infinite, and three things that finite-dimensional linear algebra gives for free have to be earned: limits of approximating sequences need not exist, linear maps need not be continuous, and bounded sets need not have convergent subsequences. This book is about what replaces each of those lost conveniences. It is written for a reader working alone, at the level of a first graduate course: the main theorems are proved in full, and each is set beside an example showing what goes wrong when one of its hypotheses is dropped.

Three courses are assumed. From Linear Algebra: vector spaces, bases, linear maps, kernels and ranges, eigenvalues, and orthogonal projection in \(\mathbb R^n\) . From Real Analysis: metric spaces, open and closed sets, compactness, uniform convergence, and the completeness of \(\mathbb R\) . From Measure Theory: the Lebesgue integral and the spaces \(L^p\) with the inequalities of Hölder and Minkowski. When a result from one of those courses is used, it is stated with the hypotheses the argument needs and marked once with its library identifier; it is not proved again. A reader who has not met \(L^p\) can still follow nearly everything by reading the sequence spaces \(\ell^p\) wherever \(L^p\) appears.

Five working skills are the goal, each to be exercised with the book closed. Decide whether a given normed space is complete and whether a given linear map is bounded, and compute or estimate its norm from both sides. State and prove the Hahn–Banach theorem, the uniform boundedness principle, and the open mapping and closed graph theorems, and say which of them a given problem calls for. Distinguish norm, weak, and weak-star convergence and extract convergent subsequences where the theory allows. Compute orthogonal projections and Riesz representers in a Hilbert space, and verify the hypotheses of the Lax–Milgram theorem for a concrete form. Recognise a compact operator, apply the fixed-point theorems of Banach, Schauder, and Kakutani, and diagonalise a compact self-adjoint operator.

Notation is fixed throughout. The scalar field \(\mathbb K\) is \(\mathbb R\) or \(\mathbb C\) . In a normed space \(X\) , the closed unit ball is \(B_X=\{x:\lVert x\rVert\le 1\}\) and \(B(x,r)\) is the open ball of radius \(r\) about \(x\) . Sequences of vectors carry a superscript or a plain index, \(x^{(n)}\) or \(x_n\) , and the \(k\) -th coordinate of a sequence \(x\) is \(x_k\) . The vector \(e_k\) has \(1\) in position \(k\) and \(0\) elsewhere. The space of bounded linear operators from \(X\) to \(Y\) is \(\mathcal B(X,Y)\) , the dual space is \(X^*\) , and inner products \(\langle x,y\rangle\) are linear in the first argument and conjugate-linear in the second.

Each chapter ends with exercises that run from computation to proof. The companion solutions give complete written proofs for some, graduated hints for others, and short answers for a few. Several exercises present a plausible argument and ask where it fails; treat them as seriously as the proofs, because the characteristic error in this subject is to apply a true theorem in a space where one of its hypotheses is false. The cumulative review at the end mixes the chapters without labels, since recognising which theorem a problem needs is most of the work.