Method of continuity


In the mathematics of Banach spaces, the method of continuity provides sufficient conditions for deducing the invertibility of one bounded linear operator from that of another, related operator.

Formulation

Let B be a Banach space, V a normed vector space, and a norm continuous family of bounded linear operators from B into V. Assume that there exists a constant C such that for every and every
Then is surjective if and only if is surjective as well.

Applications

The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations.

Proof

We assume that is surjective and show that is surjective as well.
Subdividing the interval we may assume that. Furthermore, the surjectivity of implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that is a closed subspace.
Assume that is a proper subspace. Riesz's lemma shows that there exists a such that and. Now for some and by the hypothesis. Therefore
which is a contradiction since.