Page 243 - DMTH401_REAL ANALYSIS

Unit 19: Arzelà’s Theorem and Weierstrass Approximation Theorem

Recall from Linear Algebra that the inner product space is finite-dimensional vector space V Notes
equipped with mapping ·,·: V  V  (or ), satisfying these three properties.

19.5 Review Questions

1. The Arzela-Ascoli Theorem is the key to the following result: A subset F of C(X) is compact
if and only if it is closed, bounded, and equicontinuous. Prove this.
2. You can think of Rn as (real-valued) C(X) where X is a set containing n points, and the
metric on X is the discrete metric (the distance between any two different points is 1). The
metric thus induced on Rn is equivalent to, but (unless n = 1) not the same as, the Euclidean
one, and a subset of Rn is bounded in the usual Euclidean way if and only if it is bounded
in this C(X). Show that every bounded subset of this C(X) is equicontinuous, thus
establishing the Bolzano-Weierstrass theorem as a generalization of the Arzela-Ascoli
Theorem.
3. Let f(x) = x on [0, 1] and let {f } ¥ be as in Ex. 2.3. We get Fourier coefficients a = x, f 
n n 1 n n
=
2 n
1 æ 1 ö 2 1
= x (x)dx =f - ç è 2 ø 2 = - 2 n 1 , (computation of the integral is left as exercise). Therefore,
0 ò
n ÷
+
1
we can compute Fourier series for f(x) = x which is -å ¥ f (x) .
n 1 n 1 n
=
+
2
Answer: Self Assessment
f
1. Bernstein polynomials 2. B  f uniformly
n
2
3. f = f,f = a ò b f(x) dx. 4. Orthogonal System
2
19.6 Further Readings

Books Walter Rudin: Principles of Mathematical Analysis (3rd edition), Ch. 2, Ch. 3.
(3.1-3.12), Ch. 6 (6.1 - 6.22), Ch.7(7.1 - 7.27), Ch. 8 (8.1- 8.5, 8.17 - 8.22).

G.F. Simmons: Introduction to Topology and Modern Analysis, Ch. 2(9-13),
Appendix 1, p. 337-338.
Shanti Narayan: A Course of Mathematical Analysis, 4.81-4.86, 9.1-9.9, Ch.10,Ch.14,
Ch.15(15.2, 15.3, 15.4)
T.M. Apostol: Mathematical Analysis, (2nd Edition) 7.30 and 7.31.
S.C. Malik: Mathematical Analysis.

H.L. Royden: Real Analysis, Ch. 3, 4.

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