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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==
Let ''B'' be a [[Banach space]], ''V'' a [[normed vector space]], and <math>(L_t)_{t\in[0,1]}</math> a [[Operator norm|norm]] continuous family of bounded linear operators from ''B'' into ''V''. Assume that there exists a constant ''C'' such that for every <math>t\in [0,1]</math> and every <math>x\in B</math>
:<math>||x||_B \leq C ||L_t(x)||_V.</math>
Then <math>L_0</math> is surjective if and only if <math>L_1</math> is surjective as well.


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==Applications==
The method of continuity is used in conjunction with ''a priori estimates'' to prove the existence of suitably regular solutions to [[Elliptic operator|elliptic]] [[partial differential equations]].
 
==Proof==
We assume that <math>L_0</math> is surjective and show that <math>L_1</math> is surjective as well.
 
Subdividing the interval [0,1] we may assume that <math>||L_0-L_1|| \leq 1/(3C)</math>. Furthermore, the surjectivity of <math>L_0</math> implies that ''V'' is isomorphic to ''B'' and thus a Banach space. The hypothesis implies that <math>L_1(B) \subseteq V</math> is a closed subspace.
 
Assume that <math>L_1(B) \subseteq V</math> is a proper subspace. The [[Hahn–Banach theorem]] shows that there exists a <math>y\in V</math> such that <math>||y||_V \leq 1</math> and <math>\mathrm{dist}(y,L_1(B))>2/3</math>. Now <math>y=L_0(x)</math> for some <math>x\in B</math> and <math>||x||_B \leq C ||y||_V</math> by the hypothesis. Therefore
:<math>||y-L_1(x)||_V = ||(L_0-L_1)(x)||_V \leq  ||L_0-L_1|| ||x||_B \leq 1/3,</math>
which is a contradiction since <math>L_1(x) \in L_1(B)</math>.
 
==See also==
*[[Schauder estimates]]
 
==Sources==
*{{citation|first1=D.|last=Gilbarg|first2=Neil|last2=Trudinger|authorlink2=Neil Trudinger|title=Elliptic Partial Differential Equations of Second Order|publisher=Springer|publication-place=New York|year=1983|isbn=3-540-41160-7}}
 
[[Category:Banach spaces]]
 
 
{{mathanalysis-stub}}

Revision as of 20:16, 17 September 2013

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 (Lt)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 xB

||x||BC||Lt(x)||V.

Then L0 is surjective if and only if L1 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 L0 is surjective and show that L1 is surjective as well.

Subdividing the interval [0,1] we may assume that ||L0L1||1/(3C). Furthermore, the surjectivity of L0 implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that L1(B)V is a closed subspace.

Assume that L1(B)V is a proper subspace. The Hahn–Banach theorem shows that there exists a yV such that ||y||V1 and dist(y,L1(B))>2/3. Now y=L0(x) for some xB and ||x||BC||y||V by the hypothesis. Therefore

||yL1(x)||V=||(L0L1)(x)||V||L0L1||||x||B1/3,

which is a contradiction since L1(x)L1(B).

See also

Sources

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