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 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 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 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.
Read more about this topic: Interpolation Space
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