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.
Let B be a Banach space, V a normed vector space, and ( L t ) t ∈ [ 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 ∈ [ 0 , 1 ] and every x ∈ B
| | x | | B ≤ C | | L t ( x ) | | V . Then L 0 is surjective if and only if L 1 is surjective as well.
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.
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 | | ≤ 1 / ( 3 C ) . 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 ) ⊆ V is a closed subspace.
Assume that L 1 ( B ) ⊆ V is a proper subspace. The Hahn–Banach theorem shows that there exists a y ∈ V such that | | y | | V ≤ 1 and d i s t ( y , L 1 ( B ) ) > 2 / 3 . Now y = L 0 ( x ) for some x ∈ B and | | x | | B ≤ C | | y | | V by the hypothesis. Therefore
| | y − L 1 ( x ) | | V = | | ( L 0 − L 1 ) ( x ) | | V ≤ | | L 0 − L 1 | | | | x | | B ≤ 1 / 3 , which is a contradiction since L 1 ( x ) ∈ L 1 ( B ) .
