Vol. 73, No. 1, 1977

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A general Rudin-Carlson theorem in Banach-spaces

Walter Roth

Vol. 73 (1977), No. 1, 197–213

Let K be a closed subspace in a real or complex normed linear space L. The “Main Interpolation Problem” as formulated by L. Asimow reads as follows: Given a bounded convex neighborhood V of 0 in L and a bounded closed convex U containing 0, their polars V 0 and U0 in the dual Lof L, define the functionals on LpV K(x) = sup(x,V 0 K0) and pU(x) = sup(x,U0). For x0 L we are looking for an element x L satisfying

  1. x x0 K(x|K0 = x0|K0) and
  2. pU(x) = pV K(x0) (exact solution), respectively
  3. pU(x) pV K(x0) + 𝜖 for given 𝜖 > 0 (approximate solution).

The problem is formulated in a different but equivalent way in this paper using the canonical projection p from L to L∕K. For a real linear subspace M of L, a convex cone N in M and bounded closed convex neighborhoods U and V we prove conditions in terms of the dual space of L which are necessary and sufficient for the inclusions

p(N ∩ U) ⊃ p(M )∩ p(V) resp. p(N ∩ U) ⊃ p(M )∩ p(V)

({} means the topological interior, {}, the closure).

Mathematical Subject Classification 2000
Primary: 46B99
Received: 15 February 1974
Revised: 2 June 1977
Published: 1 November 1977
Walter Roth