Interpolation space

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 $(L^{p_0}, L^{p_1})$.