Editing Compact operator (section)

== Equivalent formulations ==
A linear map <math>T: X \to Y</math> between two [[topological vector space]]s is said to be '''compact''' if there exists a neighborhood ''<math>U</math>'' of the origin in ''<math>X</math>'' such that <math>T(U)</math> is a relatively compact subset of ''<math>Y</math>''.{{sfn | Schaefer|Wolff| 1999 | p=98}}

Let <math>X,Y</math> be normed spaces and <math>T: X \to Y</math> a linear operator. Then the following statements are equivalent, and some of them are used as the principal definition by different authors<ref name=":0">{{Cite book|last=Brézis|first=H.|url=https://www.worldcat.org/oclc/695395895|title=Functional analysis, Sobolev spaces and partial differential equations|date=2011|publisher=Springer|others=H.. Brézis|isbn=978-0-387-70914-7|location=New York|oclc=695395895}}</ref>
* ''<math>T</math>'' is a compact operator;
* the image of the unit ball of ''<math>X</math>'' under ''<math>T</math>'' is [[relatively compact]] in ''<math>Y</math>'';
* the image of any bounded subset of ''<math>X</math>'' under ''<math>T</math>'' is [[relatively compact]] in ''<math>Y</math>'';
* there exists a [[neighbourhood (mathematics)|neighbourhood]] <math>U</math> of the origin in ''<math>X</math>'' and a compact subset <math>V\subseteq Y</math> such that <math>T(U)\subseteq V</math>;
* for any bounded sequence <math>(x_n)_{n\in \N}</math> in ''<math>X</math>'', the sequence <math>(Tx_n)_{n\in\N}</math> contains a converging subsequence.
If in addition ''<math>Y</math>'' is Banach, these statements are also equivalent to:
* the image of any bounded subset of ''<math>X</math>'' under ''<math>T</math>'' is [[totally bounded space|totally bounded]] in <math>Y</math>.

If a linear operator is compact, then it is continuous.