Closed range theorem

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.

Contents 1 History 2 Statement 3 Corollaries 4 References History The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.

Statement Let {displaystyle X} and {displaystyle Y} be Banach spaces, {displaystyle T:D(T)to Y} a closed linear operator whose domain {displaystyle D(T)} is dense in {displaystyle X,} and {displaystyle T'} the transpose of {displaystyle T} . The theorem asserts that the following conditions are equivalent: {displaystyle R(T),} the range of {displaystyle T,} is closed in {displaystyle Y.} {displaystyle R(T'),} the range of {displaystyle T',} is closed in {displaystyle X',} the dual of {displaystyle X.} {displaystyle R(T)=N(T')^{perp }=left{yin Y:langle x^{*},yrangle =0quad {text{for all}}quad x^{*}in N(T')right}.} {displaystyle R(T')=N(T)^{perp }=left{x^{*}in X':langle x^{*},yrangle =0quad {text{for all}}quad yin N(T)right}.} Where {displaystyle N(T)} and {displaystyle N(T')} are the null space of {displaystyle T} and {displaystyle T'} , respectively.

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

References Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11. Yosida, K. (1980), Functional Analysis, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.), Berlin, New York: Springer-Verlag. show vte Banach space topics show vte Functional analysis (topics – glossary) Categories: Banach spacesTheorems in functional analysis