Functional Analysis (6): Bounded Linear Operators and the Big Theorems

Mon, 11 Oct 2021 09:00:00 +0000

Why This Article Is Where the Theory Catches Fire#

For five articles I have been building scaffolding: metric and normed spaces, Hilbert spaces, dual spaces, weak topologies. None of those individually felt very impressive — I am, after all, just doing topology and linear algebra in slightly more general settings than usual. The point at which functional analysis genuinely delivers is right here, in the three great theorems of Banach space operator theory: the Uniform Boundedness Principle, the Open Mapping Theorem, and the Closed Graph Theorem. Each of these takes a piece of “pointwise” or “set-theoretic” data — pointwise boundedness, surjectivity, closedness of the graph — and concludes a global structural property — uniform boundedness, openness, continuity — that has no analog in finite dimensions because finite-dimensional linear algebra makes them all true automatically.

Open-Mapping on Chen Kai Blog

Functional Analysis (6): Bounded Linear Operators and the Big Theorems

Why This Article Is Where the Theory Catches Fire#