Linear Functional - Properties

Properties

  • Any linear functional L is either trivial (equal to 0 everywhere) or surjective onto the scalar field. Indeed, this follows since just as the image of a vector subspace under a linear transformation is a subspace, so is the image of V under L. But the only subspaces (i.e., k-subspaces) of k are {0} and k itself.
  • A linear functional is continuous if and only if its kernel is closed (Rudin 1991, Theorem 1.18).
  • Linear functionals with the same kernel are proportional.
  • The absolute value of any linear functional is a seminorm on its vector space.

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