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 [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be Banach spaces, [math]\displaystyle{ T : D(T) \to Y }[/math] a closed linear operator whose domain [math]\displaystyle{ D(T) }[/math] is dense in [math]\displaystyle{ X, }[/math] and [math]\displaystyle{ T' }[/math] the transpose of [math]\displaystyle{ T }[/math]. The theorem asserts that the following conditions are equivalent:
- [math]\displaystyle{ R(T), }[/math] the range of [math]\displaystyle{ T, }[/math] is closed in [math]\displaystyle{ Y. }[/math]
- [math]\displaystyle{ R(T'), }[/math] the range of [math]\displaystyle{ T', }[/math] is closed in [math]\displaystyle{ X', }[/math] the dual of [math]\displaystyle{ X. }[/math]
- [math]\displaystyle{ R(T) = N(T')^\perp = \left\{ y \in Y : \langle x^*,y \rangle = 0 \quad {\text{for all}}\quad x^* \in N(T') \right\}. }[/math]
- [math]\displaystyle{ R(T') = N(T)^\perp = \left\{x^* \in X' : \langle x^*,y \rangle = 0 \quad {\text{for all}}\quad y \in N(T) \right\}. }[/math]
Where [math]\displaystyle{ N(T) }[/math] and [math]\displaystyle{ N(T') }[/math] are the null space of [math]\displaystyle{ T }[/math] and [math]\displaystyle{ T' }[/math], respectively.
Corollaries
Several corollaries are immediate from the theorem. For instance, a densely defined closed operator [math]\displaystyle{ T }[/math] as above has [math]\displaystyle{ R(T) = Y }[/math] if and only if the transpose [math]\displaystyle{ T' }[/math] has a continuous inverse. Similarly, [math]\displaystyle{ R(T') = X' }[/math] if and only if [math]\displaystyle{ T }[/math] has a continuous inverse.
References
- Template:Banach Théorie des Opérations Linéaires
- Yosida, K. (1980), Functional Analysis, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.), Berlin, New York: Springer-Verlag.
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