TITLE: Complement of $c_{0}$ in $\ell^{\infty}$
QUESTION [19 upvotes]: How can I show that $c_{0}$ cannot be complemented in $\ell^{\infty}$?
Complement in the following sense
$$c_{0}+V = \ell^{\infty}$$
And the projections are continuous.

REPLY [7 votes]: Here is my proof.  The weak topology in the space $l^\infty$ is realcompact.  The weak topology in the space $l^\infty/c_0$ is not realcompact.  Therefore $l^\infty/c_0$ is not isomorphic to a closed subspace of $l^\infty$.  
Reference for the "weak realcompact" material:
Corson, H. H. The weak topology of a Banach space.  Trans. Amer. Math. Soc. 101 1961 1–15.
Edgar, G. A. Measurability in a Banach space. II. Indiana Univ. Math. J. 28 (1979), no. 4, 559–579.  
I showed this to W. B. Johnson once, he thought for a minute or so and came up with the more conventional proof.