Transient recovery voltage

2013-12-04T16:23:00Z

116.88.107.140:

In [[mathematics]], [[bounded function]]s are [[function (mathematics)|functions]] for which there exists a [[lower bound]] and an [[upper bound]], in other words, a constant which is larger than the [[absolute value]] of any value of this function. If we consider a [[Family (disambiguation)#Mathematics|family]] of bounded functions, this constant can vary across functions in the family. If it is possible to find one constant which bounds all functions, this family of functions is '''uniformly bounded'''.

The [[uniform boundedness principle]] in [[functional analysis]] provides sufficient conditions for uniform boundedness of a family of operators.

== Definition ==

=== Real line and complex plane ===

Let
:<math>\mathcal F=\{f_i: X \to K, i \in I\}</math>
be a family of functions [[Index set|indexed]] by <math>I</math>, where <math>X</math> is an arbitrary set and <math>K</math> is the set of [[real number|real]] or [[complex number]]s. We call <math>\mathcal F</math> '''uniformly bounded''' if there exists a real number <math>M</math> such that
:<math>|f_i(x)|\leq M \qquad \forall i \in I \quad \forall x \in X.</math>

=== Metric space ===

In general let <math>Y</math> be a [[metric space]] with metric <math>d</math>, then the set
:<math>\mathcal F=\{f_i: X \to Y, i\in I\}</math>
is called '''uniformly bounded''' if there exists an element <math>a</math> from <math>Y</math> and a real number <math>M</math> such that
:<math>d(f_i(x), a) \leq M \qquad \forall i \in I \quad \forall x \in X.</math>

==Examples==
* Every [[uniformly convergent]] sequence of bounded functions is uniformly bounded.

* The family of functions <math>f_n(x)=\sin nx\,</math> defined for [[real number|real]] <math>x</math> with <math>n</math> traveling through the [[integer]]s, is uniformly bounded by 1.

* The family of [[derivative]]s of the above family, <math>f'_n(x)=n\, \cos nx,</math> is ''not'' uniformly bounded. Each <math>f'_n\,</math> is bounded by <math>|n|,\,</math> but there is no real number <math>M</math> such that <math>|n|\le M</math> for all integers <math>n.</math>

==References==

*{{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, important to look up the site on its preface
| page = 620pp
}}

[[Category:Mathematical analysis]]

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Transient recovery voltage