| dbp:mathStatement
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- Suppose that is a real or complex vector space. If is a non-empty additive collection of balanced and absorbing subsets of then is a neighborhood base at for a vector topology on That is, the assumptions are that is a filter base that satisfies the following conditions:
# Every is balanced and absorbing,
# is additive: For every there exists a such that
If satisfies the above two conditions but is a filter base then it will form a neighborhood basis at for a vector topology on (en)
- If is a topological vector space then there exists a set of neighborhood strings in that is directed downward and such that the set of all knots of all strings in is a neighborhood basis at the origin for Such a collection of strings is said to be .
Conversely, if is a vector space and if is a collection of strings in that is directed downward, then the set of all knots of all strings in forms a neighborhood basis at the origin for a vector topology on In this case, this topology is denoted by and it is called the topology generated by (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)
- 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 topological vector space then the following four conditions are equivalent:
# The origin is closed in and there is a countable basis of neighborhoods at the origin in
# is metrizable .
# There is a translation-invariant metric on that induces on the topology which is the given topology on
# is a metrizable topological vector space.
By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant. (en)
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