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Concept# Method of continuity

Summary

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 (L_t)_{t\in[0,1]} 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
:||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|

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