Topological vector space

A topological vector space is a vector space on which in addition to its algebraic also defined yet so compatible topological structure.

Be. A vector space is a topological space at the same time, is called a topological vector space if the following compatibility axioms hold:

• The addition is continuous,
• The scalar multiplication is continuous.

Remarks:

• Sometimes it is also required in addition that a Kolmogorov space is so different points are always topologically distinguishable. It follows for topological vector spaces already the Hausdorf fig stem.
• Is the topological vector space is a Hausdorff space, so are the pictures that represent a shift by a certain vector or a stretching by a scalar, homeomorphisms. In this case it is sufficient to consider topological properties of the space in origin, since each set can be homeomorphic moved to the origin.
• Is a topological group.
• It is important that the above two pictures are not just component-wise continuous.

Examples

• The most important examples are the normed vector spaces, including the Banach spaces. More general examples are the locally convex spaces, including the Fréchet spaces.

• The quantity is a vector space, which is for the metric is a topological vector space, which is not locally convex.

• More generally be a measure space and. Then the metric makes the LP vector space to a topological space, which is not locally convex in general. Is and the counting measure, we obtain the above example. The room does not have another continuous linear functional except the zero functional.

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

Topological properties

• Every topological vector space is a uniform space as abelian topological group. He is always a particular R0 - space and satisfies the separation axiom T3 ( in the meaning, that is T0 not included ). By means of this uniform structure, one can define completeness and uniform continuity. Every topological vector space can be completed and linear continuous maps between topological vector spaces are uniformly continuous.

• For a topological vector space: T0 T1 T2 is a Tychonoff space.

• Every topological vector space has a base of neighborhoods from completed and balanced quantities.

• A topological Hausdorff shear vector space has exactly then a different constant from the zero functional linear functional, if it has a base of neighborhoods of convex sets. This fact allows to establish a rich duality theory for locally convex spaces, which does not apply for general topological vector spaces in this form. In the extreme case, see example above, the zero functional is the only continuous linear functional.