Question

Let (X, 11. I be a normed vector space and let E C X be an n-dimensional subspace. (a) Prove that E is complete. (b) Prove that E is closed. (c) Prove that dim E* = n, where E* is the algebraic dual of E (the space of all linear functionals on E).

Answer 1

Date : Solution & Let (X, 111) be a normed veeton pure and let Ecx be an n-dimensional Shabspule ofX. ^) Eis Complek: 5 Recull the theorem: het E be an n-olimensional Vector Spure , thin There is a bijection song from E to Rn. 10 Carmeny Sequence is 15 Now Let Oln be E, we Want to prove that Tans is Cauchy in Rn, het Eyo be given and let S = { yo, ITI Note that NT170, Since Tl1=0 if and only If To which is not a bejection) since (14) is Camery in E, there enists an NEN Such that if mnZN 11 n-amil 8= 20 ITU So it mm en we have 1 Inn Folm Il e Uil 12 nelmall ellTils litla UTU 25 =E So Thing is caneny in Rh But IR" is complete

Converges to Some So Orinnar lim Ilu -yll=0 y E IR", that is 5 n-D Since I s Such that a bijeckis, Loge R" Mege euusta XEE 01-Tlly cinal: 10 no ว่า 3 % lim 1124-a1l = lin Iltban) - I'ly) // lim HIT "HI Tan an - gt = 11T'll lim liTon yll zo Therefore in converges to a in Cauny Sequence in in { converges in E iso e Complete Gence every 15 in 2.

10 we b) Eis closed; het nuEx , and -(2n) be a Sequence in E, want to prove that not E, Since Olu) is a convergent Sequence in (X, 11), it is canny in Gloww ("Every convergent Sey were is Cauny) E this from a, (Alm) converges to Somene* in E il clin Xu= 2* Also Sinie Unique N=2 CE Home - E is closed in any) 15 a well limit is 20 I e) dim E*=n Ea;- algebraic dual of E het B-10,ez, ..., en? he a basis for E, then Consider the lineus functionals es defined by et eg - Sy- f o las Then we claim. That Ba=ler et ... ? forma 7 25

5 Basis for E * ;):*, et ent] is linearly 2 Indepenolert COA+60+.+ (nent=0, the Assume that Zer laras functional 10 + C2 ezt + lnen) (all=0 Vact E with X= nie,+ X2 C2+...tanen (cet +Geot + Gentleira X=1,2,..., ge* @0)+c e(eu) t...t Cuent ceu) =o 15 + =) 0+0f---+0+ Cexei* cep) +0+... 20 +0=0 Gox1=0 a=0 to Ki-12. By dehwin &*(euze And etleg -0,12 25

7 15 1..., din 2 20 4 11 B Spans Ea Let ff Ek Jun for any ne E Storeflekter) (Smile As Wet Smie n=140, + fluen for some sealers Exiflei) (Sime to linews fleret () Cimie Wizki (2) E se fa 8 flei) ei for all fe E * Hence. B spans E and dincerly independent. There for Bor from a basis for Ex And Thesky dim Eth 2) A A 25 11