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 most important examples are the normalized vector spaces , including the Banach spaces . Important concrete examples here are the Euclidean vector space , the spaces (with ) and Sobolev spaces . ${\displaystyle L^{p}}$ ${\displaystyle 1\leq p\leq \infty }$

• More general examples are the locally convex spaces , including the Fréchet spaces . Important concrete examples here are the areas of distribution theory , ie , , , , and . ${\displaystyle {\mathcal {D}}\left(\Omega \right)}$ ${\displaystyle {\mathcal {S}}\left(\Omega \right)}$ ${\displaystyle {\mathcal {E}}\left(\Omega \right)}$ ${\displaystyle {\mathcal {E}}'\left(\Omega \right)}$ ${\displaystyle {\mathcal {S}}'\left(\Omega \right)}$ ${\displaystyle {\mathcal {D}}'\left(\Omega \right)}$

• The set is a vector space that becomes a topological vector space for the metric that is not locally convex. ${\displaystyle \textstyle \ell ^{p}:=\{(x_{n})_{n}\in {\mathbb {K} }^{\mathbb {N} };\,\sum _{n=1}^{\infty }|x_{n}|^{p}<\infty \}}$ ${\displaystyle 0<p<1}$ ${\displaystyle \textstyle d_{p}((x_{n})_{n},(y_{n})_{n}):=\sum _{n=1}^{\infty }|x_{n}-y_{n}|^{p}}$

• 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 . ${\displaystyle (X,\mu )}$ ${\displaystyle 0<p<1}$ ${\displaystyle \textstyle d_{p}(f,g):=\int _{X}|f(x)-g(x)|^{p}\mathrm {d} \mu (x)}$ ${\displaystyle L^{p}(X,\mu )}$ ${\displaystyle X={\mathbb {N} }}$ ${\displaystyle \mu }$ ${\displaystyle \ell ^{p}}$ ${\displaystyle L^{p}([0,1])}$

• Each vector space has the chaotic topology , that is, only the empty set and the entire space are open , a topological vector space.

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. ${\displaystyle L^{p}([0,1])}$

See also

• Topological algebra

literature

• Helmut H. Schaefer: Topological Vector Spaces . Springer, New York et al. 1971, ISBN 0-387-98726-6 .
• Hans Jarchow: Locally Convex Spaces . Teubner, Stuttgart 1981, ISBN 3-519-02224-9 .