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| ``functions on a fin. dim. vector space''
by rmilson on 2002-06-09 00:36:03 |
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| In regards to the point raised in correction number 4.
Let's say V is an n-dimensional, real vector space.
1) Note that V is not a banach space, although it could be made
into one in an infinite number of ways.
2) It is true that V is isomorphic to R^n, in the sense
that there exist linear bijections between V and R^n,
however there is no way to prefer any one such
bijection over another.
We would like to define what it means for a
function f:V -> R to be differentiable, but how
to proceed?
We need a norm for the denominator of our limit
expression, but which norm are we to use?
The fact of the matter is that if f:V->R is differentiable
with respect to one norm, it is differentiable with
respect to all norms. This is an interesting consequence
of the finite-dimensionality of V.
Thus the concept of "differentiable function"
makes sense in the context of "finite dimensional
vector spaces", which is not quite the same context as
"Banach spaces".
A similar phenomenon occurs if we try to topologize
V. The way to proceed is to pick a norm, any norm
and to use that particular norm topology. It doesn't
matter which norm we choose to do this, we get the
same topology regardless. This is, of course,
not true in the infinite-dimensional
setting. |
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