Introduction to Functional Analysis: MIT 18.102 (Kehle, 2025)

These official MIT 18.102 notes by Christoph Kehle were prepared for Spring 2025. All 76 physical pages were read and rendered, including the seven problem sets. The archived PDF is byte-identical to the MIT copy downloaded for this ingestion; SHA-256 is a1ec1ae997c6150e00c8d3370a025b528cefc044ac0dba5ada10017cab2b9935.

Coverage and use #

The notes cover topology and metric spaces, Baire category, normed and Banach spaces, bounded operators, Hahn–Banach, uniform boundedness, open mapping, closed graph, measure and $L^p$ theory, weak and weak-* topologies, Banach–Alaoglu, reflexivity, Hilbert projection and representation, orthonormal bases, dual and Hilbert adjoints, spectrum, compact operators, the compact self-adjoint spectral theorem, and Fredholm theory. Exercises add concrete integral operators, the Volterra operator, unbounded differentiation, and a spectral solution of a Fredholm integral equation.

GTK uses the notes primarily as an independent hypothesis check. In particular:

• uniform boundedness needs a Banach domain, while the codomain may be normed;
• open mapping and bounded inverse use Banach domain and codomain;
• closed graph applies to an everywhere-defined closed linear operator between Banach spaces, not to every closed densely defined operator;
• weak-* compactness is topological compactness; sequential compactness needs additional metrizability/separability conditions;
• $L^p$ duality behaves cleanly for $1<p<\infty$, while endpoint cases require separate hypotheses and $(L^\infty)^*$ is larger than $L^1$;
• compact-operator Fredholm statements exclude the parameter $\lambda=0$.

Correction recorded during audit #

The proof following Theorem 6.21 establishes $\sigma(T)\subseteq\overline{W(T)}$, where $W(T)$ is the numerical range, but the displayed theorem omits the closure. The numerical range need not be closed for an arbitrary bounded operator. GTK uses the corrected containment and does not cite this line for a stronger claim.

Visual audit #

The document contains no separate raster or vector figures. All pages are typographic prose, formulas, proofs, and exercises; the full-page render audit found no visual that should be retained. GTK's diagrams are independent explanatory syntheses rather than copies from these notes.