Uniform boundedness

Let

:$\backslash mathcal\; F=\backslash \{f\_i:\; X\; \backslash to\; \backslash mathbb\{K\},\; i\; \backslash in\; I\backslash \}$

be a family of functions indexed by $I$, where $X$ is an arbitrary set and $\backslash mathbb\{K\}$ is either the set of real $\backslash mathbb\{R\}$ or complex numbers $\backslash mathbb\{C\}$. We call $\backslash mathcal\; F$ uniformly bounded if there exists a real number $M>0$ such that

:$|f\_i(x)|\backslash le\; M\; \backslash \; ,\; \backslash qquad\; \backslash forall\; i\; \backslash in\; I\; \backslash \; ,\; \backslash quad\; \backslash forall\; x\; \backslash in\; X.$

Another way of stating this would be the following:

:$\backslash sup\backslash limits\_\{i\; \backslash in\; I\}\; \backslash sup\backslash limits\_\{x\; \backslash in\; X\}\; |f\_i(x)|\backslash le\; M.$

In general let $Y$ be a metric space with metric $d$, then the set

:$\backslash mathcal\; F=\backslash \{f\_i:\; X\; \backslash to\; Y,\; i\backslash in\; I\backslash \}$

is called uniformly bounded if there exists an element $a$ from $Y$ and a real number $M$ such that

:$d(f\_i(x),\; a)\; \backslash leq\; M\; \backslash qquad\; \backslash forall\; i\; \backslash in\; I\; \backslash quad\; \backslash forall\; x\; \backslash in\; X.$

• Every uniformly convergent sequence of bounded functions is uniformly bounded.
• The family of functions $f\_n(x)=\backslash sin\; nx$ defined for real $x$ with $n$ traveling through the integers, is uniformly bounded by 1.
• The family of derivatives of the above family, $f\text{'}\_n(x)=n\backslash ,\; \backslash cos\; nx,$ is not uniformly bounded. Each $f\text{'}\_n$ is bounded by $|n|,$ but there is no real number $M$ such that $|n|\backslash le\; M$ for all integers $n.$

• {{cite book

| last = Ma

| first = Tsoy-Wo

| title = Banach–Hilbert spaces, vector measures, group representations

| publisher = World Scientific

| date = 2002

| isbn = 981-238-038-8

| page = 620pp

}}

Category:Mathematical analysis