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.

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 positive constant C such that for every

is surjective if and only if

is surjective as well.

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.

We assume that

is surjective and show that

is surjective as well.

Subdividing the interval [0,1] 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

{\displaystyle \mathrm {dist} (y,L_{1}(B))>2/3}

by the hypothesis.

Therefore which is a contradiction since