Question

Show that every finite dimensional norm linear space is complete

Answer 1

Let V be a vector space over a complete topological field say R (or C) with dim(V)=n, base ei and norm ‖⋅‖. Let vk be a Cauchy sequence w.r.t. ‖⋅‖. Since any two norms on a finite dimensional space are equivalent, ‖⋅‖ is equivalent to the l1-norm ‖⋅‖1 which means that for some constant C, ε>0, k,j large enough,ε>‖vj−vk‖≥C‖vj−vk‖1ei=Cn∑i=1|vji−vki|≥|vji−vki|for each 1≤i≤n. Hence vki is a Cauchy sequence in R (or C) for each i. R (or C) is complete hence vi=limk→∞vki is in R (or C) for each i. Let v=(v1,…,vn)=∑iviei. Then v is in V and ‖vk−v‖→0:

Let ε>0. Then‖vk−v‖≤C‖vk−v‖1=Cn∑i=1|vki−vi|≤C′nε

for k large enough.