u8910520的部落格

since Translations and multiplication
by a nonzero scalar are homeophisms on C[0,1]
so It`s suffices to show closure of B(0,2) isn`t compact
(why?)

consider
fn(x)=nx if x belong to [0,1/n]
=1 if x belong to [1/n,1]

let eplison=1/2,for any delta>0
there exist n is natural number s.t 1/n < 1/2..
then d(1/n,0)< delta
but |fn(1/n)-fn(0)|=1 >1/2..

so we get {fn} isn`t equiconti
by Arzela Ascoli Thm
closure of B(0,2) isn`t compact.
so C[0,1] isn`t locally compact. QED