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FK-space

From Wikipedia, the free encyclopedia

In functional analysis and related areas of mathematics a FK-space or Fréchet coordinate space is a sequence space equipped with a topological structure such that it becomes a Fréchet space. FK-spaces with a normable topology are called BK-spaces.

There exists only one topology to turn a sequence space into a Fréchet space, namely the topology of pointwise convergence. Thus the name coordinate space because a sequence in an FK-space converges if and only if it converges for each coordinate.

FK-spaces are examples of topological vector spaces. They are important in summability theory.

Definition

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A FK-space is a sequence space of , that is a linear subspace of vector space of all complex valued sequences, equipped with the topology of pointwise convergence.

We write the elements of as with .

Then sequence in converges to some point if it converges pointwise for each That is if for all

Examples

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The sequence space of all complex valued sequences is trivially an FK-space.

Properties

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Given an FK-space of and with the topology of pointwise convergence the inclusion map is a continuous function.

FK-space constructions

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Given a countable family of FK-spaces with a countable family of seminorms, we define and Then is again an FK-space.

See also

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References

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