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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 functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
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.)
A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the Borsuk–Ulam theorem says that a continuous map from the n-dimensional sphere to R n has a pair of antipodal points that are mapped to the same point.
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.
The following sets will constitute the basic open subsets of topologies on spaces of linear maps. For any subsets and , let (,):= {: ()}.. The family {(,):,} forms a neighborhood basis [1] at the origin for a unique translation-invariant topology on , where this topology is not necessarily a vector topology (that is, it might not make into a TVS).