## Are there any significant differences between studying functional analysis from a normed space perspective versus a metric space perspective?

Does it matter if functional analysis was introduced from a normed space versus a metric space formulation? Are all major theorems from functional analysis (such as Banach contraction mapping, Hahn Banach theorem…) directly transferable if we exchanged every instance of norm with metric? I am asking because strictly speaking a metric is different than a … Read more