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.

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Formulation

Let B be a Banach space, V a normed vector space, and (L_t)_{t\in[0,1]} a norm continuous family of bounded linear operators from B into V. Assume that there exists a constant C such that for every t\in [0,1] and every x\in B

||x||_B \leq C ||L_t(x)||_V.

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 ||L_0-L_1|| \leq 1/(3C). Furthermore, the surjectivity of L_0 implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that L_1(B) \subseteq V is a closed subspace.

Assume that L_1(B) \subseteq V is a proper subspace. The Hahn-Banach theorem shows that there exists a y\in V such that ||y||_V \leq 1 and \mathrm{dist}(y,L_1(B))>2/3. Now y=L_0(x) for some x\in B and ||x||_B \leq C ||y||_V by the hypothesis. Therefore

||y-L_1(x)||_V = ||(L_0-L_1)(x)||_V \leq ||L_0-L_1|| ||x||_B \leq 1/3,

which is a contradiction since L_1(x) \in L_1(B).

See also

Sources