Proposition 14

Let X be a locally compact topological space, $\mu$ a positive measure on X, and $p\in [1,+\infty ]$. The space $L^p_K(X, \mu)$ is an approximation space.

If $p= +\infty$, then the space $L^{\infty}_{\mathbf{C}}(X, \mu)$ is a commutative stellar algebra (Example 4 of I, p. 103), hence an approximation space (Cor. of Prop. 13), and the same is true of $L^{\infty}_{\mathbf{R}}(X, \mu)$ by Remark 5 of III, p. 15.

Suppose that $p$ is finite. Let $A\subset \mathscr{L}^f(L^p_K(X, \mu))$ be the set of projectors of finite rank and norm $\leqslant 1$. By Lemma 6, the identity mapping of $L^p_K(X, \mu)$ is adherent to A for the topology of simple convergence, and the proposition then follows from Remark 3 of III, p. 14.