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 ${\displaystyle (L_{t})_{t\in [0,1]}}$ a norm continuous family of bounded linear operators from B into V. Assume that there exists a positive constant C such that for every ${\displaystyle t\in [0,1]}$ and every ${\displaystyle x\in B}$

${\displaystyle ||x||_{B}\leq C||L_{t}(x)||_{V}.}$

Then ${\displaystyle L_{0}}$ is surjective if and only if ${\displaystyle L_{1}}$ 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 ${\displaystyle L_{0}}$ is surjective and show that ${\displaystyle L_{1}}$ 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 ${\displaystyle L_{0}}$ 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 ${\displaystyle x\in B}$ 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