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FK space

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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.

Contents

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

A FK-space is a sequence space X , 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 X as

( x n ) n ∈ N with x n ∈ C

Then sequence ( a n ) n ∈ N ( k ) in X converges to some point ( x n ) n ∈ N if it converges pointwise for each n . That is

lim k → ∞ ( a n ) n ∈ N ( k ) = ( x n ) n ∈ N

if

∀ n ∈ N : lim k → ∞ a n ( k ) = x n

Examples

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

    • Properties

    Given an FK-space X and ω with the topology of pointwise convergence the inclusion map

    ι : X → ω

    is continuous.

    FK-space constructions

    Given a countable family of FK-spaces ( X n , P n ) with P n a countable family of semi-norms, we define

    X := ⋂ n = 1 ∞ X n

    and

    P := { p | X ∣ p ∈ P n } .

    Then ( X , P ) is again an FK-space.

    References

    FK-space Wikipedia
    (Text) CC BY-SA