|
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. == Properties == A continuous linear operator maps bounded sets into bounded sets. A linear functional is continuous if and only if its kernel is closed. Every linear function on a finite-dimensional space is continuous. The following are equivalent: given a linear operator ''A'' between topological spaces ''X'' and ''Y'': # ''A'' is continuous at 0 in ''X''. # ''A'' is continuous at some point in ''X''. # ''A'' is continuous everywhere in ''X''. The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality : for any set ''D'' in ''Y'' and any ''x''0 in ''X'', which is true due to the additivity of ''A''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「continuous linear operator」の詳細全文を読む スポンサード リンク
|