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|
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading