Interpolation Space - Duality

Duality

Let (X₀, X₁) be a compatible couple, and assume that X₀ ∩ X₁ is dense in X₀ and in X₁. In this case, the restriction map from the (continuous) dual X ′_j of X_j, j = 0, 1, to the dual of X₀ ∩ X₁ is one-to-one. It follows that the pair of duals (X ′₀, X ′₁) is a compatible couple continuously embedded in the dual (X₀ ∩ X₁) ′.

For the complex interpolation method, the following duality result holds:

Theorem. Let (X₀, X₁) be a compatible couple of complex Banach spaces, and assume that X₀ ∩ X₁ is dense in X₀ and in X₁. If X₀ and X₁ are reflexive, then the dual of the complex interpolation space is obtained by interpolating the duals,

In general, the dual of the space (X₀, X₁)_θ is equal to (X ′₀, X ′₁)θ, a space defined by a variant of the complex method. The upper-θ and lower-θ methods do not coincide in general, but they do for reflexive spaces.

For the real interpolation method, the duality holds provided that the parameter q is finite:

Theorem. Let 0 < θ < 1, 1 ≤ q < ∞ and (X₀, X₁) a compatible couple of real Banach spaces. Assume that X₀ ∩ X₁ is dense in X₀ and in X₁. Then