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 positive constant C such that for every t\in [0,1] and every x\in B Then L_0 is surjective if and only if L_1 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 L_0 is surjective and show that L_1 is surjective as well. Subdividing the interval [0,1] we may assume that. Furthermore, the surjectivity of L_0 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 y\in V such that and. Now y=L_0(x) for some x\in B and by the hypothesis. Therefore which is a contradiction since.

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