1.3. Convergence

1.3.1. Pointwise convergence

Suppose \(\{f_n\}\) be a sequence of functions sharing the same domain \(X\) and codomain \(Y\).

The sequence \(\{f_n\}\) converges pointwise to \(f\), written as:

\[\lim_{n \to \infty} f_n = f\]

if and only if

\[\lim_{n \to \infty} f_n(x) = f(x) \quad \forall x \in X\]

1.3.2. Uniform convergence

A sequence \(\{f_n\}\) converges uniformly to \(f\), if the speed of convergence of \(\{f_n(x)\}\) to \(f(x)\) doesn’t depend upon \(x\).

Consider the sequence \(a_n = \sup_x|f_n(x) - f(x)|\) where the supremum is taken over all \(x \in X\). \(\{f_n\}\) converges uniformly to \(f\) if and only if \(a_n \to 0\).

• Uniform convergence is stronger than pointwise convergence.

• Every uniformly convergent sequence is pointwise convergent.

1.3.3. Almost everywhere convergence

In measure theory, the concept of almost everywhere convergence of a sequence of measurable functions defined on a measurable space is discussed. It means pointwise convergence almost everywhere (i.e. the measure of points at which convergence doesn’t happen is zero).

1.3.4. Absolute convergence

An infinite series of numbers is said to converge absolutely if the sum of absolute values of the summand is finite.

\(\sum_{-\infty}^{+\infty}a_n\) is said to converge absolutely if \(\sum_{-\infty}^{+\infty}|a_n| < \infty\).

This can also be defined for a one sided series \(\sum_{0}^{+\infty}a_n\).

Similarly an improper integral of a function \(\int_{-\infty}^{\infty} f(x)dx\) is said to converge absolutely if \(\int_{-\infty}^{\infty} |f(x)|dx < \infty\); i.e. the integral of the absolute value of the integrand is finite.

• Any absolutely convergent series is convergent.

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$Id: convergence.rst 249 2012-08-05 06:17:57Z shailesh $