# Topological vector space

A topological vector space is a vector space on which not only its algebraic but also a compatible topological structure is defined.

## definition

Be . A - vector space , which also topological space is called a topological vector space if the following compatibility axioms: ${\ displaystyle \ mathbb {K} \ in \ {\ mathbb {R}, \ mathbb {C} \}}$ ${\ displaystyle \ mathbb {K}}$ ${\ displaystyle E}$ • The vector addition is continuous ,${\ displaystyle E \ times E \ to E}$ • The scalar multiplication is continuous.${\ displaystyle \ mathbb {K} \ times E \ to E}$ ## Remarks

• It is important that the two maps mentioned are not only continuous in terms of components.
• Sometimes it is also required that a Kolmogoroff space (i.e. T 0 space) is, i.e. that different points are always topologically distinguishable. The Hausdorff property (i.e. T 2 space) already follows for topological vector spaces .${\ displaystyle E}$ • ${\ displaystyle (E, +)}$ is a topological group .
• For a topological vector space , the topological dual space can be explained in a meaningful way .${\ displaystyle E}$ ${\ displaystyle E '}$ • If the topological vector space is a Hausdorff space, then the mappings that represent a shift by a certain vector or a stretching by a scalar are homeomorphisms . In this case it is sufficient to consider topological properties of the space in the origin, since any amount can be moved homeomorphically into the origin.

## Examples

• The set is a vector space that becomes a topological vector space for the metric that is not locally convex.${\ displaystyle \ textstyle \ ell ^ {p}: = \ {(x_ {n}) _ {n} \ in {\ mathbb {K}} ^ {\ mathbb {N}}; \, \ sum _ {n = 1} ^ {\ infty} | x_ {n} | ^ {p} <\ infty \}}$ ${\ displaystyle 0 ${\ displaystyle \ textstyle d_ {p} ((x_ {n}) _ {n}, (y_ {n}) _ {n}): = \ sum _ {n = 1} ^ {\ infty} | x_ { n} -y_ {n} | ^ {p}}$ • More general are a measure space and . Then the metric makes the L p space a topological vector space that is generally not locally convex. If and is the counting measure , the above example is obtained . Apart from the zero functional, space has no other continuous linear functional .${\ displaystyle (X, \ mu)}$ ${\ displaystyle 0 ${\ displaystyle \ textstyle d_ {p} (f, g): = \ int _ {X} | f (x) -g (x) | ^ {p} \ mathrm {d} \ mu (x)}$ ${\ displaystyle L ^ {p} (X, \ mu)}$ ${\ displaystyle X = {\ mathbb {N}}}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ ell ^ {p}}$ ${\ displaystyle L ^ {p} ([0,1])}$ ## Topological properties

• As an Abelian, topological group, every topological vector space is a uniform space . In particular, it is always an R 0 space and fulfills the axiom of separation T 3 (meaning that T 0 is not included). This uniform structure can be used to define completeness and uniform continuity . Any topological vector space can be completed and linear continuous mappings between topological vector spaces are uniformly continuous.
• For a topological vector space is considered: is T 0 is T 1 is T 2 is a Tychonoff space .${\ displaystyle X}$ ${\ displaystyle X}$ ${\ displaystyle \ Leftrightarrow}$ ${\ displaystyle X}$ ${\ displaystyle \ Leftrightarrow}$ ${\ displaystyle X}$ ${\ displaystyle \ Leftrightarrow}$ ${\ displaystyle X}$ • Every topological vector space has a zero neighborhood basis made up of closed and balanced sets . According to Kolmogoroff's criterion for normalization , a Hausdorff topological vector space can be normalized if and only if it has a bounded and convex null neighborhood.
• In locally convex Hausdorff topological vector spaces which applies Hahn-Banach theorem , so the existence of "many" continuous linear functional is secured. This fact makes it possible to set up a rich duality theory for such spaces , which does not apply to general topological vector spaces in this form. In the extreme case, as in the example above , the zero functional is the only continuous linear functional.${\ displaystyle L ^ {p} ([0,1])}$ 