190 CHAPTER 8. NORMED LINEAR SPACES

The vectors,{

u j}n

j=1 , generated in this way are therefore an orthonormal basis becauseeach vector has unit length.

The process by which these vectors were generated is called the Gram Schmidt process.

8.4 Equivalence Of NormsAs mentioned above, it makes absolutely no difference which norm you decide to use. Thisholds in general finite dimensional normed spaces. First are some simple lemmas featuringone dimensional considerations. In this case, the distance is given by d (x,y) = |x− y| andso the open balls are sets of the form (x−δ ,x+δ ).

Also recall the Theorem 3.0.3 which is stated next for convenience.

Lemma 8.4.1 The closed interval [a,b] is sequentially compact.

Corollary 8.4.2 The set Q≡ [a,b]+ i [c,d]⊆ C is compact, meaning

{x+ iy : x ∈ [a,b] ,y ∈ [c,d]}

Proof: Let {xn + iyn} be a sequence in Q. Then there is a subsequence such that

limk→∞

xnk = x ∈ [a,b] .

There is a further subsequence such that liml→∞ ynkl= y ∈ [c,d]. Thus, also

liml→∞

xnkl= x

because subsequences of convergent sequences converge to the same point. Therefore,from the way we measure the distance in C, it follows that liml→∞

(xnkl

+ ynkl

)= x+ iy ∈

Q.The next corollary gives the definition of a closed disk and shows that, like a closed

interval, a closed disk is compact.

Corollary 8.4.3 In C, let D(z,r)≡ {w ∈ C : |z−w| ≤ r}. Then D(z,r) is compact.

Proof: Note that

D(z,r)⊆ [Rez− r,Rez+ r]+ i [Imz− r, Imz+ r]

which was just shown to be compact. Also, if wk → w where wk ∈ D(z,r) , then by thetriangle inequality,

|z−w|= limk→∞

|z−wk| ≤ r

and so D(z,r) is a closed subset of a compact set. Hence it is compact by Proposition 7.6.8.

Recall that sequentially compact and compact are the same in any metric space whichis the context of the assertions here.