Editing Compact-open topology (section)

== Fréchet differentiable functions ==
Let {{mvar|X}} and {{mvar|Y}} be two [[Banach space]]s defined over the same [[field (mathematics)|field]], and let {{math|''C<sup>&thinsp;m</sup>''(''U'', ''Y'')}} denote the set of all {{mvar|m}}-continuously [[Fréchet derivative|Fréchet-differentiable]] functions from the open subset {{math|''U'' ⊆ ''X''}} to {{mvar|Y}}. The compact-open topology is the [[initial topology]] induced by the [[seminorm]]s

:<math>p_{K}(f) = \sup \left\{ \left\| D^j f(x) \right\| \ : \ x \in K, 0 \leq j \leq m \right\}</math>

where {{math|''D''<sup>0</sup>&thinsp;''f''&thinsp;(''x'') {{=}} &thinsp;''f''&thinsp;(''x'')}}, for each compact subset {{math|''K'' ⊆ ''U''}}.{{clarification needed|date=February 2022|reason=Is this original research showing that this definition is equivalent in this special case to the general definition given above? Or is it a definition copied from an external reference, in which case that reference should be cited?}}