Closed range theorem

In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range.

History

The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.

Statement

Let $X$ and $Y$ be Banach spaces, $T\colon D(T)\to Y$ a closed linear operator whose domain $D(T)$ is dense in $X$ , and $T'$ the transpose of $T$ . The theorem asserts that the following conditions are equivalent:

$R(T)$ , the range of $T$ , is closed in $Y$ ,
$R(T')$ , the range of $T'$ , is closed in $X'$ , the dual of $X$ ,
$R(T)=N(T')^{\perp }=\{y\in Y|\langle x^{*},y\rangle =0\quad {{\text{for all}}}\quad x^{*}\in N(T')\}$ ,
$R(T')=N(T)^{\perp }=\{x^{*}\in X'|\langle x^{*},y\rangle =0\quad {{\text{for all}}}\quad y\in N(T)\}$ .

Corollaries

Several corollaries are immediate from the theorem. For instance, a densely defined closed operator $T$ as above has $R(T)=Y$ if and only if the transpose $T'$ has a continuous inverse. Similarly, $R(T')=X'$ if and only if $T$ has a continuous inverse.

References

Yosida, K. (1980), Functional Analysis, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.), Berlin, New York: Springer-Verlag .

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

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–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

This article is issued from Wikipedia - version of the 11/29/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Closed range theorem

History

Statement

Corollaries

See also

References