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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 ...
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.
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.)
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).
In particular, every linear functional of a finite-dimensional Hausdorff TVS is continuous. Finite-dimensional range : Any continuous surjective linear map L : X → Y {\displaystyle L:X\to Y} with a Hausdorff finite-dimensional range is an open map [ 1 ] and thus a topological homomorphism .
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.
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. [1] [2] [3] That is, a function : is open if for any open set in , the image is open in . Likewise, a closed map is a function that maps closed sets to closed sets.