Kenneth Kuttler

122 CHAPTER 4. LINEAR SPACES

Show X \V is a dense open subset of X which is equivalent to V containing no ballB(v,r) ,{w : ∥w− v∥< r}. Hint: If B(x,r) is contained in V, then show, that since Vis a subspace, B(0,r) is contained in V. Then show this implies X ⊆ V which is notthe case.

12. Show that if (X ,d) is a metric space and H,K are disjoint closed sets, there areopen sets UH ,UK such that H ⊆ UH ,K ⊆ UK and UH ∩UK = /0. Hint: Let k ∈K. Explain why dist(k,H) ≡ inf{∥k−h∥ : h ∈ H} ≡ 2δ k > 0. Now consider UK ≡∪k∈KB(k,δ k). Do something similar for h ∈ H and consider UH ≡ ∪k∈HB(h,δ h).

13. If, in a metric space, B(p,δ ) is a ball, show that

B(p,δ )⊆ D(p,δ )≡ {x : ∥x− p∥ ≤ δ}

Now suppose (X ,d) is a complete metric space and Un,n∈N is a dense open set in X .Also let W be any nonempty open set. Show there exists a ball B1 ≡ B(p1,r1) havingradius smaller than 2−1 such that B1⊆U1∩W1. Next show there exists B2≡B(p2,r2)such that B2 ⊆ B1 ∩U2 ∩W with the radius of B2 less than 2−2. Continue this way.Explain why {pn}∞

n=1 is a Cauchy sequence converging to some p ∈W ∩ (∪∞n=1Un).

This is the very important Baire theorem which says that in a complete metric space,the intersection of dense open sets is dense.

14. Suppose you have a complete normed linear space, (X ,∥·∥). Use the above problemsleading to the Baire theorem in 13 to show that if B is a Hamel basis for for X , thenB cannot be countable. Hint: If B = {vi}∞

i=1 , consider Vn ≡ span(v1, ...,vn) . Thenuse a problem listed above to argue that VC

n is a dense open set. Now apply Problem13. This shows why the idea of a Hamel basis often fails to be very useful whereas,in finite dimensional settings, it is just what is needed.

15. In any complete normed linear space which is infinite dimensional, show the unitball is not compact. Do this by showing the existence of a sequence which cannothave a convergent subsequence. Hint: Pick ∥x1∥ = 1. Suppose x1, ...,xn have beenchosen, each ∥xk∥= 1. Then there is x /∈ span(x1, ...,xn)≡Vn. Now consider v suchthat ∥x− v∥ ≤ 3

2 dist(x,Vn) . Then argue that for k ≤ n,

∥∥∥∥ x− v∥x− v∥

− xk

∥∥∥∥=∥∥∥∥∥∥∥∥∥∥∥∥∥∥

x−

 ∈Vn︷︸︸︷v+∥x− v∥xk

∥x− v∥

∥∥∥∥∥∥∥∥∥∥∥∥∥∥≥ dist(x,Vn)

(3/2)dist(x,Vn)=

16. Let X be a complete inner product space. Let F denote subsets β ⊆ X such thatwhenever x,y ∈ X ,(x,y) = 0 if x ̸= y and (x,x) = 1 if x = y. Thus these β are or-thonormal sets. Show there exists a maximal orthonormal set. If X is separable, showthat this maximal orthonormal set is countable. Hint: Use the Hausdorff maximaltheorem. The next few problems involve linear algebra.

17. Let X be a real inner product space and let {v1, ...,vn} be vectors in X . Let G be then×n matrix Gi j ≡ (vi,v j) . Show that G−1 exists if and only if {v1, ...,vn} is linearlyindependent. G is called the Grammian or the metric tensor.

12212.13.14.15.16.17.CHAPTER 4. LINEAR SPACESShow X \V is a dense open subset of X which is equivalent to V containing no ballB(v,r),{w: ||w—v]| <r}. Hint: If B (x,r) is contained in V, then show, that since Vis a subspace, B(0,r) is contained in V. Then show this implies X C V which is notthe case.Show that if (X,d) is a metric space and H,K are disjoint closed sets, there areopen sets Uy,Ux such that H C Uy,K C Ux and UyNUx =9. Hint: Let k €K. Explain why dist (k,H) = inf {||k—Al| :h € H} = 26; > 0. Now consider Ux =UxexB (k, 6). Do something similar for A € H and consider Uy = UgerB (h, 6).If, in a metric space, B(p, 6) is a ball, show thatB(p.8) CD(p,6) = {x: ||x— pl <5}Now suppose (X,d) is a complete metric space and U,,,n € N is a dense open set in X.Also let W be any nonempty open set. Show there exists a ball Bj = B(p1,1r1) havingradius smaller than 2~! such that B} C U; NW. Next show there exists By = B(p2,12)such that By C Bj NU2 MW with the radius of By less than 2~2. Continue this way.Explain why {p,};_, is a Cauchy sequence converging to some p € WM (Ur_,U,).This is the very important Baire theorem which says that in a complete metric space,the intersection of dense open sets is dense.Suppose you have a complete normed linear space, (X, ||-||). Use the above problemsleading to the Baire theorem in 13 to show that if @ is a Hamel basis for for X, thenZ cannot be countable. Hint: If Z = {v;}* , , consider V, = span (v1,...,¥,). Thenuse a problem listed above to argue that V,© is a dense open set. Now apply Problem13. This shows why the idea of a Hamel basis often fails to be very useful whereas,in finite dimensional settings, it is just what is needed.In any complete normed linear space which is infinite dimensional, show the unitball is not compact. Do this by showing the existence of a sequence which cannothave a convergent subsequence. Hint: Pick ||a || = 1. Suppose x1,...,x, have beenchosen, each ||x;|| = 1. Then there is x ¢ span (x1,...,X) = Vn. Now consider v suchthat ||x —v|| < 3 dist (x,V,). Then argue that for k <n,EVnOOx— | vt |[x—v|| xgdist (x, V;) 2xXx—vV\|x — v|| = (3/2)dist(x,V,) 3ATV oyl|x—v|Let X be a complete inner product space. Let Y denote subsets B C X such thatwhenever x,y € X,(x,y) =0 if x # y and (x,x) = 1 if x= y. Thus these B are or-thonormal sets. Show there exists a maximal orthonormal set. If X is separable, showthat this maximal orthonormal set is countable. Hint: Use the Hausdorff maximaltheorem. The next few problems involve linear algebra.Let X be a real inner product space and let {v1,...,v,} be vectors in X. Let G be thenxn matrix Gj; = (vj,v;). Show that G~! exists if and only if {v1,..., Vn} is linearlyindependent. G is called the Grammian or the metric tensor.