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[NP] More on Duality Pair Structure: Closed Range Theorem

NP Originally published 2020-01-20

For a Hilbert space, we have inner product, so we have orthogonal complement; as we mensioned on the preceding post, for Banach space (only with norm), we can use duality pair to construct annihilator. This induces that the following are equivalent:
  • Range of T is closed;
  • Range of T is the annihilator of the null space of T';
  • Range of T' is closed;
  • Range of T' is the annihilator of the null space of T;