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 \( {\displaystyle (L_{t})_{t\in [0,1]}} \) a norm continuous family of bounded linear operators from B into V. Assume that there exists a constant C such that for every \( t\in [0,1] \) and every \( x\in B \)
\( {\displaystyle ||x||_{B}\leq C||L_{t}(x)||_{V}.} \)
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 \( {\displaystyle ||L_{0}-L_{1}||\leq 1/(3C)} \). Furthermore, the surjectivity of \( L_{0} \) implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that \( {\displaystyle L_{1}(B)\subseteq V} \) is a closed subspace.
Assume that \( {\displaystyle L_{1}(B)\subseteq V} \) is a proper subspace. Riesz's lemma shows that there exists a \({\displaystyle y\in V} \) such that \( {\displaystyle ||y||_{V}\leq 1} \) and \( {\displaystyle \mathrm {dist} (y,L_{1}(B))>2/3} \). Now \( {\displaystyle y=L_{0}(x)} \) for some \( x\in B \) and \( {\displaystyle ||x||_{B}\leq C||y||_{V}} \) by the hypothesis. Therefore
\({\displaystyle ||y-L_{1}(x)||_{V}=||(L_{0}-L_{1})(x)||_{V}\leq ||L_{0}-L_{1}||||x||_{B}\leq 1/3,} \)
which is a contradiction since \( {\displaystyle L_{1}(x)\in L_{1}(B)}. \)
See also
Schauder estimates
Sources
Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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