Method of continuity

Short description: Procedure for determining if an operator is invertible

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.

## Contents

* 1 Formulation
* 2 Applications
* 3 Proof
* 4 See also
* 5 Sources

## Formulation

Let B be a Banach space, V a normed vector space, and [math]\displaystyle{ (L_t)_{t\in[0,1]} }[/math] a norm continuous family of bounded linear operators from B into V. Assume that there exists a constant C such that for every [math]\displaystyle{ t\in [0,1] }[/math] and every [math]\displaystyle{ x\in B }[/math]

[math]\displaystyle{ ||x||_B \leq C ||L_t(x)||_V. }[/math]

Then [math]\displaystyle{ L_0 }[/math] is surjective if and only if [math]\displaystyle{ L_1 }[/math] 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 [math]\displaystyle{ L_0 }[/math] is surjective and show that [math]\displaystyle{ L_1 }[/math] is surjective as well.

Subdividing the interval [0,1] we may assume that [math]\displaystyle{ ||L_0-L_1|| \leq 1/(3C) }[/math]. Furthermore, the surjectivity of [math]\displaystyle{ L_0 }[/math] implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that [math]\displaystyle{ L_1(B) \subseteq V }[/math] is a closed subspace.

Assume that [math]\displaystyle{ L_1(B) \subseteq V }[/math] is a proper subspace. Riesz's lemma shows that there exists a [math]\displaystyle{ y\in V }[/math] such that [math]\displaystyle{ ||y||_V \leq 1 }[/math] and [math]\displaystyle{ \mathrm{dist}(y,L_1(B))\gt 2/3 }[/math]. Now [math]\displaystyle{ y=L_0(x) }[/math] for some [math]\displaystyle{ x\in B }[/math] and [math]\displaystyle{ ||x||_B \leq C ||y||_V }[/math] by the hypothesis. Therefore

[math]\displaystyle{ ||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]\displaystyle{ L_1(x) \in L_1(B) }[/math].

## 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

* v
* t
* e

Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

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