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Jun 22, 2018 at 11:07 vote accept Bernhard
Jun 22, 2018 at 10:33 comment added Jochen Wengenroth What you write is okay (and quite elementary) but I don't think that this is usually called Grothendieck's completeness theorem. There is a theorem of Grothendieck about the completion of a locally convex space (the space of linear functional on the continuous dual with weak$^*$-continuous restrictions to all equi-continuous sets endowed with the topology of uniform convergence on those sets is a completion).
Jun 22, 2018 at 9:45 history answered Parschallen CC BY-SA 4.0