154 CHAPTER 5. FUNCTIONS ON NORMED LINEAR SPACES
∥g− fn∥→ 0. This is the second Müntz theorem. Hint: Show that you can approxi-mate x→ xm uniformly. To do this, use the above Müntz to approximate mxm−1 with∑k ckxpk−1 in the inner product norm.
∫ 10
∣∣mxm−1−∑nk=1 ckxpk−1
∣∣2 dx ≤ ε2. Thenxm−∑
nk=1
ckpk
xpk =∫ x
0(mtm−1−∑
nk=1 ckt pk−1
)dt. Then∣∣∣∣∣xm−
n
∑k=1
ck
pkxpk
∣∣∣∣∣≤∫ x
0
∣∣∣∣∣mtm−1−n
∑k=1
ckt pk−1
∣∣∣∣∣dt ≤∫ 1
01
∣∣∣∣∣mtm−1−n
∑k=1
ckt pk−1
∣∣∣∣∣dt
Now use the Cauchy Schwarz inequality on that last integral to obtain
maxx∈[0,1]
∣∣∣∣∣xm−n
∑k=1
ck
pkxpk
∣∣∣∣∣≤ ε.
In case m = 0, there is nothing to show because 1 is in Vn. Explain why the resultfollows from this and the Weierstrass approximation theorem.