- Art Gallery -

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 $${\displaystyle (L_{t})_{t\in [0,1]}}$$ 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 $$x\in B$$

$${\displaystyle ||x||_{B}\leq C||L_{t}(x)||_{V}.}$$

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 $${\displaystyle ||L_{0}-L_{1}||\leq 1/(3C)}$$. Furthermore, the surjectivity of $$L_{0}$$ implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that $${\displaystyle L_{1}(B)\subseteq V}$$ is a closed subspace.

Assume that $${\displaystyle L_{1}(B)\subseteq V}$$ is a proper subspace. Riesz's lemma shows that there exists a $${\displaystyle y\in V}$$ such that $${\displaystyle ||y||_{V}\leq 1}$$ and $${\displaystyle \mathrm {dist} (y,L_{1}(B))>2/3}$$. Now $${\displaystyle y=L_{0}(x)}$$ for some $$x\in B$$ and $${\displaystyle ||x||_{B}\leq C||y||_{V}}$$ by the hypothesis. Therefore

$${\displaystyle ||y-L_{1}(x)||_{V}=||(L_{0}-L_{1})(x)||_{V}\leq ||L_{0}-L_{1}||||x||_{B}\leq 1/3,}$$

which is a contradiction since $${\displaystyle L_{1}(x)\in L_{1}(B)}.$$

Schauder estimates

Sources
Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7