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 a norm continuous family of bounded linear operators from B into V. Assume that there exists a positive constant C such that for every and every
Then is surjective if and only if is surjective as well.
Applicationsedit
The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to ellipticpartial differential equations.
Proofedit
We assume that is surjective and show that is surjective as well.
Subdividing the interval [0,1] we may assume that . Furthermore, the surjectivity of implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that is a closed subspace.
Assume that is a proper subspace. Riesz's lemma shows that there exists a such that and . Now for some and by the hypothesis. Therefore
Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN3-540-41160-7
January 01, 1970
method, continuity, mathematics, banach, spaces, method, continuity, provides, sufficient, conditions, deducing, invertibility, bounded, linear, operator, from, that, another, related, operator, contents, formulation, applications, proof, also, sourcesformulat. 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 Contents 1 Formulation 2 Applications 3 Proof 4 See also 5 SourcesFormulation editLet B be a Banach space V a normed vector space and L t t 0 1 displaystyle L t t in 0 1 nbsp 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 0 1 displaystyle t in 0 1 nbsp and every x B displaystyle x in B nbsp x B C L t x V displaystyle x B leq C L t x V nbsp Then L 0 displaystyle L 0 nbsp is surjective if and only if L 1 displaystyle L 1 nbsp is surjective as well Applications editThe 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 editWe assume that L 0 displaystyle L 0 nbsp is surjective and show that L 1 displaystyle L 1 nbsp is surjective as well Subdividing the interval 0 1 we may assume that L 0 L 1 1 3 C displaystyle L 0 L 1 leq 1 3C nbsp Furthermore the surjectivity of L 0 displaystyle L 0 nbsp implies that V is isomorphic to B and thus a Banach space The hypothesis implies that L 1 B V displaystyle L 1 B subseteq V nbsp is a closed subspace Assume that L 1 B V displaystyle L 1 B subseteq V nbsp is a proper subspace Riesz s lemma shows that there exists a y V displaystyle y in V nbsp such that y V 1 displaystyle y V leq 1 nbsp and d i s t y L 1 B gt 2 3 displaystyle mathrm dist y L 1 B gt 2 3 nbsp Now y L 0 x displaystyle y L 0 x nbsp for some x B displaystyle x in B nbsp and x B C y V displaystyle x B leq C y V nbsp by the hypothesis Therefore y L 1 x V L 0 L 1 x V L 0 L 1 x B 1 3 displaystyle y L 1 x V L 0 L 1 x V leq L 0 L 1 x B leq 1 3 nbsp which is a contradiction since L 1 x L 1 B displaystyle L 1 x in L 1 B nbsp See also editSchauder estimatesSources editGilbarg D Trudinger Neil 1983 Elliptic Partial Differential Equations of Second Order New York Springer ISBN 3 540 41160 7 Retrieved from https en wikipedia org w index php title Method of continuity amp oldid 1158794835, wikipedia, wiki, book, books, library,