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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.
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.
We write the elements of as
Then sequence in converges to some point if it converges pointwise for each . That is
- The sequence space of all complex valued sequences is trivially an FK-space.
Given an FK-space and with the topology of pointwise convergence the inclusion map
Given a countable family of FK-spaces with a countable family of semi-norms, we define
Then is again an FK-space.