Interpolation Space - The Setting of Interpolation

The Setting of Interpolation

A Banach space X is said to be continuously embedded in a Hausdorff topological vector space Z when X is a linear subspace of Z such that the inclusion map from X into Z is continuous. A compatible couple (X₀, X₁) of Banach spaces consists of two Banach spaces X₀ and X₁ that are continuously embedded in the same Hausdorff topological vector space Z. The embedding in a linear space Z allows to consider the two linear subspaces

Interpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of X₀ and X₁. It depends in an essential way from the specific relative position that X₀ and X₁ occupy in a larger space Z.

One can define norms on X₀ ∩ X₁ and X₀ + X₁ by

Equipped with these norms, the intersection and the sum are Banach spaces. The following inclusions are all continuous:

Interpolation studies the family of spaces X that are intermediate spaces between X₀ and X₁ in the sense that

where the two inclusions maps are continuous.

An example of this situation is the pair (L1(R), L∞(R)), where the two Banach spaces are continuously embedded in the space Z of measurable functions on the real line, equipped with the topology of convergence in measure. In this situation, the spaces Lp(R), for 1 ≤ p ≤ ∞ are intermediate between L1(R) and L∞(R). More generally,

with continous injections, so that, under the given condition, Lp(R) is intermediate between Lp₀(R) and Lp₁(R).

Definition. Given two compatible couples (X₀, X₁) and (Y₀, Y₁), an interpolation pair is a couple (X, Y) of Banach spaces with the two following properties:

• The space X is intermediate between X₀ and X₁, and Y is intermediate between Y₀ and Y₁.
• If L is any linear operator from X₀ + X₁ to Y₀ + Y₁, which maps continuously X₀ to Y₀ and X₁ to Y₁, then it also maps continuously X to Y.

The interpolation pair (X, Y) is said to be of exponent θ (with 0 < θ < 1) if there exists a constant C such that

for all operators L as above. The notation ||L||_{X, Y} is for the norm of L as a map from X to Y. If C = 1, one says that (X, Y) is an exact interpolation pair of exponent θ.