Method of continuity

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 $(L_t)_{t\in[0,1]}$ 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 $x\in B$

$||x||_B \leq C ||L_t(x)||_V.$

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 $||L_0-L_1|| \leq 1/(3C)$. Furthermore, the surjectivity of L₀ implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that $L_1(B) \subseteq V$ is a closed subspace.

Assume that $L_1(B) \subseteq V$ is a proper subspace. The Hahn-Banach theorem shows that there exists a $y\in V$ such that $||y||_V \leq 1$ and dist(y,L₁(B)) > 2 / 3. Now y = L₀(x) for some $x\in B$ and $||x||_B \leq C ||y||_V$ by the hypothesis. Therefore

$||y-L_1(x)||_V = ||(L_0-L_1)(x)||_V \leq ||L_0-L_1|| ||x||_B \leq 1/3,$

which is a contradiction since $L_1(x) \in L_1(B)$.

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 

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