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If : is a bounded linear operator from a normed space into some TVS then : is necessarily continuous; this is because any open ball centered at the origin in is both a bounded subset (which implies that () is bounded since is a bounded linear map) and a neighborhood of the origin in , so that is thus bounded on this neighborhood of the origin ...
Nearly/Almost open linear maps. A linear map : between two topological vector spaces (TVSs) is called a nearly open map (or sometimes, an almost open map) if for every neighborhood of the origin in the domain, the closure of its image is a neighborhood of the origin in . [18] Many authors use a different definition of "nearly/almost open ...
A sequentially continuous linear map between two TVSs is always bounded, [1] but the converse requires additional assumptions to hold (such as the domain being bornological and the codomain being locally convex). If the domain is also a sequential space, then is sequentially continuous if and only if it is continuous.
The usual proof of the closed graph theorem employs the open mapping theorem.It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see closed graph theorem § Relation to the open mapping theorem (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)
Pavel Urysohn. In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma [1]) states that any real-valued, continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
Theorem — Given a barrelled space and a locally convex space, then any family of pointwise bounded continuous linear mappings from to is equicontinuous (and even uniformly equicontinuous). Alternatively, the statement also holds whenever X {\displaystyle X} is a Baire space and Y {\displaystyle Y} is a locally convex space.
Any continuous function into a Hausdorff space has a closed graph (see § Closed graph theorem in point-set topology) Any linear map, :, between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) is sequentially continuous in the sense of the product ...
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.