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Method of continuity

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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

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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.

Applications

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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

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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

which is a contradiction since .

See also

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Sources

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