In the mathematics of

Banach spaces In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...

, the method of continuity provides sufficient conditions for deducing the invertibility of one

bounded linear operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vect ...

from that of another, related operator.

Formulation

Let ''B'' be a

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...

, ''V'' a

normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "leng ...

, and

(L_t)_

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...

continuous family of bounded linear operators from ''B'' into ''V''. Assume that there exists a positive constant ''C'' such that for every

t\in,1 /math> and every x\in B :, , 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 In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

Proof

We assume that

L_0

is surjective and show that

L_1

is surjective as well. Subdividing the interval ,1we may assume that

, , 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

L_1(B) \subseteq V

is a closed subspace. Assume that

L_1(B) \subseteq V

is a proper subspace.

Riesz's lemma Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense. The lemma may also be called the Riesz lemma or Riesz inequal ...

shows that there exists a

y\in V

such that

, , y, , _V \leq 1

and

\mathrm(y,L_1(B))>2/3

. Now

y=L_0(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)

Sources

* {{Functional analysis Banach spaces

Formulation

Applications

Proof

See also

Sources