Let SX be a closed and bounded subset of a finite-dimensional normed linear space X with basis

Let SŠ†X be a closed and bounded subset of a finite-dimensional normed linear space X with basis {x1, x2, . . . ; xn}, and let xm be a sequence in S.
Every term xm has a unique representation

1.Using lemma 1.1, show that for every i the sequence of scalars (xmi) is bounded
2.Show that (xm) has a subsequence (xm(I)) for which the coordinates of the first coordinate xm1 converge to α.
3.Repeating this argument n times, show that (xm) has a subsequence whose scalars converge to (x1,x2,...,xn).
4.Define

Show that xm †’ x.
5. Show that x ˆˆ S.
6. Conclude that is compact.
An immediate corollary is that the closed unit ball in a finite dimensional space

is compact (since it is closed and bounded). This is not the case in an infinite-dimensional space, so a linear space is finite-dimensional if and only if its closed unit ball is compact.

Transcribed Image Text:

Σ a"x; m. х" 'X х, i=l