How to check the compactness of these sets:

Question

a.the unit sphere in $l_2$ the space of all square summable real sequences with its usual metric i.e.$d({x_i,y_i}) =(\sum_1^\infty|x_i-y_i|^2)^{1/2}$

b.the closure of the unit ball.

I find many equivalent conditions on compact sets but dont know which to use

Each set contains the standard unit vectors. Does this set have a limit point? — David Mitra, Nov 22 '14 at 16:27

score 1 · Accepted Answer · answered Nov 22 '14 at 16:30

1

The closed unit ball is not compact in any infinite dimensional Banach space. Thus the closure of unit ball is not compact. If the unit sphere in $\ell_2 $ were compact then the sequence $\{e_n\}$ of vectors of orthogonal basis of $\ell_2$ should has a convergent subsequence but this is impossible since $||e_i -e_j| =\sqrt{2}.$

answered Nov 22 '14 at 16:30

How do you get the first line of your proof? – Learnmore Nov 22 '14 at 17:37
Please look here: http://math.stackexchange.com/questions/163500/an-application-of-riesz-lemma and here : http://www.math.umn.edu/~garrett/m/fun/riesz_lemma.pdf – Nov 22 '14 at 17:41

How to check the compactness of these sets:

1 Answers1