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.