2
$\begingroup$

Is this little toy known ?

Let $E$ be some Banach space, and let $K$ be the closed unit ball of its dual, endowed with the weak-star topology. Also, let $j:E$ $\rightarrow$ $C(K)$ be the natural embedding. Then, if $\pi$ :$E$ $\rightarrow$ $C(K)$ is onto, one must have $\left\Vert \pi-j\right\Vert $ > 1. [Applying this to $E=$ $\ell^{1}$, or to $E=C[0,1]$ (eventually, via Milutin) would be interesting, I think.]

Improve this question
$\endgroup$
3
  • $\begingroup$ What's up with this "subjective-argumentative" tag? I question whether it (i) applies here and (ii) would be a useful tag for any post. It seems to me that application of this tag is, itself, rather subjective and argumentative. $\endgroup$
    – Pete L. Clark
    Commented Apr 5, 2010 at 1:31
  • $\begingroup$ The remark looks too trivial to give much of anything. Note that the conclusion follows if e.g. you only assume that the range $\pi$ contains the constant functions. $\endgroup$
    – Bill Johnson
    Commented Apr 5, 2010 at 15:52
  • $\begingroup$ Maybe less trivial: it is true also when $\pi$ has a dense range. $\endgroup$
    – Ady
    Commented Apr 6, 2010 at 3:04

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .