Sequences and Series, Absolute Convergence and Norms
Absolute Convergence and Norms
If the norm of a vector function is absolutely convergent, so is the vector function.
To illustrate, let s contain complex numbers whose norms form an absolutely convergent series.
The absolute value of the real component of each term is positive,
and bounded by the norm of that term.
Thus the real components are dominated by the norms,
and they converge absolutely.
Similarly, the imaginary components converge absolutely,
hence s is absolutely convergent.
Conversely, let s be an absolutely convergent series of complex numbers.
The norm of each term is bounded by the absolute value of its real component
plus the absolute value of its imaginary component.
In other words, the norms are bounded by the sum of two absolutely convergent series,
hence the norms are absolutely convergent.