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- Vector space over a topological fieldA topological vector space (TVS) is a vector space over a topological field1. It is endowed with a topology such that vector addition and scalar multiplication are continuous functions1. The domains of these functions are endowed with product topologies1. Every TVS is a commutative topological group with identity under addition2. The canonical uniformity of a TVS is defined entirely in terms of subtraction (and thus addition); scalar multiplication is not involved and no additional structure is needed2.Learn more:✕This summary was generated using AI based on multiple online sources. To view the original source information, use the "Learn more" links.A topological vector space (TVS) is a vector space over a topological field (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that vector addition and scalar multiplication are continuous functions (where the domains of these functions are endowed with product topologies).en.wikipedia.org/wiki/Topological_vector_spaceEvery topological vector space (TVS) is a commutative topological group with identity under addition and the canonical uniformity of a TVS is defined entirely in terms of subtraction (and thus addition); scalar multiplication is not involved and no additional structure is needed.en.wikipedia.org/wiki/Complete_topological_vector…
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Topological vector space - Wikipedia
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space … See more
Normed spaces
Every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a … See moreA topological vector space (TVS) $${\displaystyle X}$$ is a vector space over a topological field $${\displaystyle \mathbb {K} }$$ (most often the real or See more
A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as … See more
Every topological vector space has a continuous dual space—the set $${\displaystyle X'}$$ of all continuous linear functionals, that is, continuous linear maps from the space into the base field $${\displaystyle \mathbb {K} .}$$ A topology on the … See more
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