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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.

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    This is called Phillips's lemma. A nice proof can be found in Robert Whitley, *[Projecting $m$ onto $c_0$](http://dx.doi.org/10.2307/2315346)*, The American Mathematical Monthly Vol. **73** (3) (Mar., 1966), pp. 285-286. But most texts on basic functional analysis contain a proof, e.g. Conway.2012-04-16
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    The original reference is Phillips, *[On linear transformations](http://dx.doi.org/10.1090/S0002-9947-1940-0004094-3)*, Trans. Amer. Math. Soc. **48** (1940), 516-541, see 7.5 on page 539.2012-04-16

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