To prove lemma 1.1, assume, to the contrary, that for every c > 0 there exists x ˆˆ lin{x1; x2; . . . , xn} such that

where x = a1x1 + a2x2 + €¢ €¢ €¢ +‡anxn. Show that this implies that
1. There exists a sequence (x''') with |x'''||†’0
2. There exists a subsequence converging to some x ˆˆ lin {x1,x2,...xn}
3. x ‰  0 contradicting the conclusion that ||x'''|| †’ 0
This contradiction proves the existence of a constant c > 0 such that ||x|| ‰¥ c(|α1| + |α2| + ... +|αn|) for every x ˆˆ lin S.

Transcribed Image Text:

i=1