Topological Vector Space
A topological vector space is a vector space that is also a topological space, where the vector space operations (addition and scalar multiplication) are continuous with respect to the topology.
e.g. Banach spaces, Hilbert spaces, and Sobolev spaces are all examples of topological vector spaces.
Locally Convex Space
In practice, it is often useful to define a locally convex space by means of a family of seminorms.
Seminorm
A seminorm on a complex or real topological vector space
is a map such that the following properties hold:
- nonnegativity:
for all , - absolute homogeneity:
for all and , - triangle inequality:
for all .
Alternative Characterization of Locally Convex Spaces
A topological vector space
is locally convex if and only if its topology is induced by a countable family of seminorms. That is, its topology is the coarsest topology such that each seminorm is continuous.
Fréchet Spaces
Fréchet Space
A topological vector space
is a Fréchet space if it is, locally convex, metrizable and complete.
Remark
The original definition due to Stefan Banach defines Fréchet spaces as metrizable complete topological vector spaces, which does not require local convexity. The books of Waelbroeck (1971) and Wilansky (1978) follow this original definition.
The following propositions help to verify if a given space is Fréchet:
Proposition
A locally convex space is metrizable if and only if its topology can be generated by a countable separating family of seminorms.
e.g.
- Clearly, any Banach space is a Fréchet space, since it is complete and metrizable by the norm.
- The space of real smooth functions
is a Fréchet space with the seminorms defined by for each nonnegative integers and . - If
is a compact smooth manifold, then the space of smooth functions is a Fréchet space. This holds for some non-compact manifolds as well, such as the real line .