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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.
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.
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.)
So, if the open mapping theorem holds for ; i.e., is an open mapping, then is continuous and then is continuous (as the composition of continuous maps). For example, the above argument applies if f {\displaystyle f} is a linear operator between Banach spaces with closed graph, or if f {\displaystyle f} is a map with closed graph between compact ...
The continuous linear functionals on B(H) for the ultraweak, ultrastrong, ultrastrong * and Arens-Mackey topologies are the same, and are the elements of the predual B(H) *. By definition, the continuous linear functionals in the norm topology are the same as those in the weak Banach space topology.
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.