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- In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS. (en)
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- 61296 (xsd:nonNegativeInteger)
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- Let be metric on a vector space such that the topology induced by on makes into a topological vector space. If is a complete metric space then is a complete-TVS. (en)
- Every complete metrizable TVS with the Hahn-Banach extension property is locally convex. (en)
- All infinite-dimensional separable complete metrizable TVS are homeomorphic. (en)
- If is an additive commutative topological group then the following are equivalent:
is induced by a pseudometric; ;
is induced by a translation-invariant pseudometric;
the identity element in has a countable neighborhood basis.
If is Hausdorff then the word "pseudometric" in the above statement may be replaced by the word "metric."
A commutative topological group is metrizable if and only if it is Hausdorff and pseudometrizable. (en)
- If is a TVS whose topology is induced by a paranorm then is complete if and only if for every sequence in if then converges in (en)
- If is a group , is a topology on and is endowed with the product topology, then the addition map is continuous at the origin of if and only if the set of neighborhoods of the origin in is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood." (en)
- Suppose that is an additive commutative group.
If is a translation invariant pseudometric on then the map is a value on called the value associated with , and moreover, generates a group topology on .
Conversely, if is a value on then the map is a translation-invariant pseudometric on and the value associated with is just (en)
- If is a topological vector space then the following three conditions are equivalent:
The origin is closed in and there is a countable basis of neighborhoods for in
is metrizable .
There is a translation-invariant metric on that induces on the topology which is the given topology on
By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant. (en)
- Suppose that is a locally convex metrizable TVS and that is a countable sequence of bounded subsets of
Then there exists a bounded subset of and a sequence of positive real numbers such that for all (en)
- If is a sequence in a locally convex metrizable TVS that converges to some then there exists a sequence in such that in and each is a convex combination of finitely many (en)
- Let be a collection of subsets of a vector space such that and for all
For all let
Define by if and otherwise let
Then is subadditive and on so in particular
If all are symmetric sets then and if all are balanced then for all scalars such that and all
If is a topological vector space and if all are neighborhoods of the origin then is continuous, where if in addition is Hausdorff and forms a basis of balanced neighborhoods of the origin in then is a metric defining the vector topology on (en)
- If is a pseudometrizable TVS whose topology is induced by a pseudometric then is a complete pseudometric on if and only if is complete as a TVS. (en)
- Let be an F-seminorm on a vector space
Then the map defined by
is a translation invariant pseudometric on that defines a vector topology on
If is an F-norm then is a metric.
When is endowed with this topology then is a continuous map on
The balanced sets as ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set.
Similarly, the balanced sets as ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of open sets. (en)
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- In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS. (en)
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- Metrizable topological vector space (en)
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