reflection shows that in limit processes and many other considerations
we use the fact that on R we have available a distance function,call
it d.which associates a distance d(x,y)=|x-y|with every pair of points
x,y€R.In the plane and in "ordinary" three-dimensional space the
situation is similar.
In functional analysis we shall study more general "spaces" and
"functions" defined on them.We arrive at a sufficiently general and
flexible concept of a "space" as follows.We replace the set of real
numbers underlying R by an abstract set X and introduce on X a "distance function"which has only a few of the most fundamental properties of the distance function on R.
But what do we mean by "most fundamental"?
This question is far from being trivial. In fact,the choice and formulation of axioms in a definition always needs experience,familiarity with practical problems and a clear idea of the goal to be reached.
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