Content deleted Content added
Fixed notation |
|||
Line 91:
since the map <math>f : [0, \infty) \to \R</math> defined by <math>f(x) = x^q</math> is a continuous non-decreasing function whose domain <math>[0, \infty)</math> always contains <math>S := \{|g(x)| : x \in \Omega\}</math> and <math>\sup S \,\stackrel{\scriptscriptstyle\text{def}}{=}\, \|g\|_\infty^q.</math>
Although this discussion focused on <math>\sup,</math> similar conclusions can be reached for <math>\inf</math> with appropriate changes (such as requiring that <math>f</math> be non-increasing rather than non-decreasing). Other [[Norm (mathematics)|norms]] defined in terms of <math>\sup</math> or <math>\inf</math> include the [[weak Lp space|weak <math>L^{p,w}</math> space]] norms (for <math>1 \leq p < \infty</math>), the norm on [[Lp space|Lebesgue space]] <math>L^\infty(\Omega, \mu),</math> and [[operator norm]]s. Monotone sequences in <math>S</math> that converge to <math>\sup S</math> (or to <math>\inf S</math>) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations.
===Arithmetic operations on sets===
| |||