Interpolation of Banach spaces

A compatible Banach couple consists of two Banach spaces continuously embedded in a common Hausdorff topological vector space. Their intersection and sum can then be formed as actual vector subspaces, without ambiguity about when two vectors are equal. The sum norm is the infimum of the endpoint costs over decompositions. For each positive t, the K-functional changes the relative price of the second component. An element can be cheap because it has a good decomposition, not because it lies in both endpoint spaces with individually small norms.

For 0<θ<1 and 1≤r≤∞, the real interpolation space collects sum-space vectors whose weighted K-functional has finite Lʳ norm for measure dt/t. When r is infinite, replace the integral by the supremum. The logarithmic measure treats different scales comparably, while θ determines their weighting. Different r values can yield different intermediate spaces even with the same θ; for Lebesgue endpoints, Lorentz spaces occur. In particular, one should not identify every real interpolation choice with a single Lᵖ space or silently treat equivalent norms as isometric identities.

Let T be a single linear map from X₀+X₁ into Y₀+Y₁, with consistent restrictions bounded from Xᵢ to Yᵢ by positive constants Mᵢ. Apply T to any decomposition of x. The two endpoint bounds give a candidate decomposition cost for Tx, and taking the infimum yields the displayed K-functional estimate. Substituting a rescaled integration variable then gives the geometric operator bound with exponents one minus θ and θ. The same scaling works for the supremum norm when r is infinite.

This proof identifies the required consistency condition: two separately defined endpoint operators must agree on the intersection before they define one operator on the sum. It also identifies the spaces controlled: real interpolation uses the same θ and r on the source and target couples. Endpoint bounds equal to zero can be handled by replacing them with arbitrarily small positive upper bounds and taking a limit. The theorem gives an upper estimate, so strict inequality in a numerical example is expected and does not signal a failure of interpolation.

For complex-valued Lᵖ spaces on sigma-finite measure spaces, suppose a consistent linear operator on the sum of two source endpoint spaces is bounded from $L^{p_0}$ to $L^{q_0}$ and from $L^{p_1}$ to $L^{q_1}$. Riesz–Thorin gives the geometric endpoint bound at reciprocal exponents lying on the connecting line. Each endpoint exponent lies between one and infinity, with reciprocal infinity interpreted as zero. Its proof uses complex interpolation and a three-lines argument. This is distinct from the real K-construction, despite the similar geometric bound and overlapping applications.

On a finite counting space, endpoint norms become induced matrix norms: the ℓ¹ operator norm is the largest absolute column sum and the ℓ∞ norm is the largest absolute row sum. Interpolating them at the midpoint bounds the ℓ² norm by their geometric mean. Singular values give that Euclidean norm exactly, allowing a concrete check of both the theorem and its possible slack. Infinite-dimensional applications still require the specified measure, exponent and operator hypotheses; a successful matrix calculation does not remove those assumptions or establish an arbitrary nonlinear interpolation principle.