Homework Help: What does this notation mean?

1. Oct 2, 2011

gauss mouse

If you look here http://planetmath.org/encyclopedia/RiezsLemma.html [Broken], there seems to be something missing - nothing is said about the norm of (x_alpha) or about the norm of (s - x_alpha).

Now, the same thing seems to happen here http://planetmath.org/encyclopedia/CompactnessOfClosedUnitBallInNormedSpaces.html [Broken], so I guess there's something about the notation that I'm not getting, rather than there being something actually missing.

Can anyone help? To be honest, I fail to see how "lx_alphal and ls-x_alphal for every s in S" could be a statement.

NB: The notation l.l is used to denote norm on the quoted webpages, rather than the more usual ll.ll

Last edited by a moderator: May 5, 2017
2. Oct 2, 2011

(Riesz Lemma). Fix 0 < $\alpha$ < 1. If S$\subset$ E is a proper closed subspace of a
Banach space E then one can find x$_{\alpha}$ $\in$ X with ||x$_{\alpha}$|| = 1 and |s - x| $\geq$  $\alpha$, for all s $\in$ S