Closed range theorem

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

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)\}.

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.

See also[edit]

References[edit]

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