---
tag: V60F20N6JT
kind: definition
name: Interpolation space
msc: [46B70]
defines: VX1R6FN6PS
uses: [06GZ956657, 03TM135G0N]
sources:
  - url: https://en.wikipedia.org/wiki/Interpolation_space
---

## Definition

Let $X_0$ and $X_1$ be [[Banach spaces|06GZ956657]] that are both continuously embedded in a Hausdorff topological vector space $Z$.
The pair $(X_0, X_1)$ is a *compatible couple*.
The spaces $X_0 \cap X_1$ and $X_0 + X_1$ are Banach spaces with the norms
$$\norm{x}_\cap = \max(\norm{x}_{X_0}, \norm{x}_{X_1}), \qquad \norm{x}_+ = \inf \set{\norm{x_0}_{X_0} + \norm{x_1}_{X_1} : x = x_0 + x_1}.$$
A Banach space $X$ is an *intermediate space* if $X_0 \cap X_1 \subseteq X \subseteq X_0 + X_1$, with both inclusions continuous.
An intermediate space $X$ is an *interpolation space* if the following holds.
Every linear map $T \colon X_0 + X_1 \to X_0 + X_1$ that restricts to [[bounded linear maps|03TM135G0N]] $X_0 \to X_0$ and $X_1 \to X_1$ maps $X$ into $X$, and the restriction $T \colon X \to X$ is bounded.

## Remarks

Example: $X_0 + X_1$ is an interpolation space.
If $x = x_0 + x_1$, then $T x = T x_0 + T x_1$ with $T x_0 \in X_0$ and $T x_1 \in X_1$.
So $\norm{T x}_+ \le \max(\norm{T}_{X_0}, \norm{T}_{X_1}) \, \norm{x}_+$.

Non-example: take $X_0 = \ell^1$ and $X_1 = \ell^\infty$, and let $X = \ell^1 \oplus \R \mathbf{1}$, where $\mathbf{1} = (1, 1, 1, \dots)$.
With the norm $\norm{y + c \mathbf{1}} = \norm{y}_1 + \abs{c}$, the space $X$ is intermediate.
The map $T(x_1, x_2, x_3, \dots) = (x_1, 0, x_2, 0, x_3, 0, \dots)$ has norm $1$ on $\ell^1$ and on $\ell^\infty$.
But $T \mathbf{1} = (1, 0, 1, 0, \dots)$ is not in $X$, since it differs from every multiple of $\mathbf{1}$ by a sequence that is not summable.

By the Riesz–Thorin theorem, for a measure space and $1 \le p_0 < p < p_1 \le \infty$, the space $L^p$ is an interpolation space for the couple $(L^{p_0}, L^{p_1})$.