Interpolation Space - Discrete Definitions

Discrete Definitions

Since the function tK(x, t) varies regularly (it is increasing, but K(x, t) / t is decreasing), the definition of the Kθ, 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 xK(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 Jθ, p-norm of a functional x' in the dual of (X0, X1)θ, q :

The usual formula for the discrete Jθ, 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.

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