Interpolation Space - Discrete Definitions

Discrete Definitions

Since the function t → K(x, t) varies regularly (it is increasing, but K(x, t) / t is decreasing), the definition of the K_θ, q-norm of a vector x, previously given by an integral, is equivalent to a definition given by a series. This series is obtained by breaking (0, ∞) into pieces (2n, 2n+1) of equal mass for the measure d t / t,

In the special case where X₀ is continuously embedded in X₁, one can omit the part of the series with negative indices n. In this case, each of the functions x → K(x, 2n; X₀, X₁) defines an equivalent norm on X₁.

The interpolation space (X₀, X₁)_θ, q is a "diagonal subspace" of an ℓq-sum of a sequence of Banach spaces (each one being isomorphic to X₀ + X₁). Therefore, when q is finite, the dual of (X₀, X₁)_θ, q  is a quotient of the ℓp-sum of the duals, 1 / p + 1 / q = 1, which leads to the following formula for the discrete J_θ, p-norm of a functional x' in the dual of (X₀, X₁)_θ, q :

The usual formula for the discrete J_θ, p-norm is obtained by changing n to −n.

The discrete definition makes several questions easier to study, among which the already mentioned identification of the dual. Other such questions are compactness or weak-compactness of linear operators. Lions and Peetre have proved that:

Theorem. If the linear operator T is compact from X₀ to a Banach space Y and bounded from X₁ to Y, then T is compact from (X₀, X₁)_θ, q to Y when 0 < θ < 1, 1 ≤ q ≤ ∞.

Davis, Figiel, Johnson and Pełczyński have used interpolation in their proof of the following result:

Theorem. A bounded linear operator between two Banach spaces is weakly compact if and only if it factors through a reflexive space.