Measure, Integration & Real Analysis · Problem 6D.14

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · banach-limit-functional

Problem

Show that there exists a linear functional

\varphi: \ell^\infty \to \mathbf{F}

such that

|\varphi(a_1, a_2, \ldots)| \leq \|(a_1, a_2, \ldots)\|_\infty

for all

(a_1, a_2, \ldots) \in \ell^\infty

and

\varphi(a_1, a_2, \ldots) = \lim_{k \to \infty} a_k

for all

(a_1, a_2, \ldots) \in \ell^\infty

such that the limit above on the right exists.