Сызықтық операторлардың үздіктігі және шектеулілігі туралы

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.

A linear map is "bounded on a neighborhood" (of some point) if and only if it is locally bounded at every point of its domain, in which case it is necessarily continuous (even if its domain is not a normed space) and thus also bounded (because a continuous linear operator is always a bounded linear operator). For any linear map, if it is bounded on a neighborhood then it is continuous, and if it is continuous then it is bounded. The converse statements are not true in general but they are both true when the linear map's domain is a normed space. Examples and additional details are now given below.

The next example shows that it is possible for a linear map to be continuous (and thus also bounded) but not bounded on any neighborhood. In particular, it demonstrates that being "bounded on a neighborhood" is not always synonymous with being "bounded". Example: A continuous and bounded linear map that is not bounded on any neighborhood: If is the identity map on some locally convex topological vector space then this linear map is always continuous (indeed, even a TVS isomorphism) and bounded, but is bounded on a neighborhood if and only if there exists a bounded neighborhood of the origin in which is equivalent to being a seminormable space (which if is Hausdorff, is the same as being a normable space). This shows that it is possible for a linear map to be continuous but not bounded on any neighborhood. Indeed, this example shows that every locally convex space that is not seminormable has a linear TVS automorphism that is not bounded on any neighborhood of any point. Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.

Every linear functional on a topological vector space (TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them. However, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.

Every linear map whose domain is a finite dimensional Hausdorff topological vector space (TVS) is continuous. This is not true if the finite dimensional TVS is not Hausdorff. Every (constant) map between TVS that is identically equal to zero is a linear map that is continuous, bounded, and bounded on the neighborhood of the origin. In particular, every TVS has a non empty continuous dual space (although it is possible for the constant zero map to be its only continuous linear functional). Suppose is any Hausdorff TVS. Then every linear functional on is necessarily continuous if and only if every vector subspace of is closed. Every linear functional on is necessarily a bounded linear functional if and only if every bounded subset of is contained in a finite dimensional vector subspace.