Baire Category and the Uniform Boundedness Principle
In a nonempty complete metric space, countably many nowhere dense sets cannot cover the space. Baire's theorem yields uniform boundedness for pointwise-bounded families of bounded linear operators on Banach spaces and proves the open mapping theorem for surjective bounded linear operators between Banach spaces.
What a learner can do afterwards
- State the category theorem and point at the step where completeness is used
- Derive a uniform bound from pointwise bounds on a family of operators
- Say what the open mapping and closed graph theorems add to the same argument
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Closure means the set together with all its limit points. An open set contains a ball around each of its points. A set is nowhere dense when its closure contains no open ball. A meagre set, also called first category, is a countable union of nowhere dense sets. The rationals are dense but meagre in the real line: they are countable, and each single point is nowhere dense.
A metric space is complete when every Cauchy sequence converges to a point in it. The rationals are incomplete: decimal truncations of sqrt(3) are rational and Cauchy but have an irrational limit. Baire says a nonempty complete metric space is not meagre in itself. Its proof uses nested closed balls with shrinking radii whose centres are Cauchy; completeness supplies a limit outside the proposed union.
Let bounded linear T_n on Banach X satisfy sup_n ||T_n x|| < infinity for each x. Closed sets E_k = {x: all ||T_n x|| <= k} cover X. Baire gives a closed ball of radius r > 0 at a in one E_k. For ||h|| <= r, both a+h and a lie there. Linearity and the triangle inequality give ||T_n h|| <= ||T_n(a+h)|| + ||T_n a|| <= 2k. Scaling gives ||T_n|| <= 2k/r. If ||T_n|| <= M, then ||T_n x|| <= M||x||; T_n = M times the identity attains this for M >= 0.
Open mapping: a surjective bounded linear T between Banach spaces is open. Baire, differences and scaling put a ball around 0 in the closure of T(B), with B the open unit ball. Approximate a target by Tx_j with ||x_j|| < 1, 1/2, 1/4, ... and halved residual bounds. Geometric tail bounds make partial sums Cauchy. Completeness gives their limit; continuity maps it to the target. Its norm is < 2, so T(2B) contains the ball. Translation and scaling prove openness.
Closed graph: a linear T between Banach spaces X and Y with closed graph G is bounded. The product X times Y has norm max(||x||,||y||); Cauchy pairs converge coordinatewise, so it is Banach. Its closed subspace G is Banach. The projection P(x,Tx) = x is bounded and bijective. Open mapping makes P^-1 bounded. The target projection Q is bounded too, so T = Q P^-1 is bounded.
Nested closed balls in Baire's proof have radii tending to 0. Late centres lie in the same small ball, so they form a Cauchy sequence. Completeness gives a limit in the space; closedness keeps it in every ball, outside the nowhere dense sets being avoided.
For bounded linear operators on a Banach space, pointwise bounds give one bound on all operator norms.
Original lesson, questions and flashcards authored for this dataset.
Original notes and questions authored for this dataset. Enrichment flashcards were drafted from collected fact candidates and edited for this topic.
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