Re: teminology



On Fri, 13 Apr 2007, fishfry wrote:

In article <1176455217.528539.209410@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"gius" <giu.giorgi@xxxxxxxxx> wrote:

Hi, everyone,

In mathematics,
what does it mean for a map defined in a certain set,
say A,
to extend "naturally" to a set B which contains as a proper subset
the set A ?


It means the "obvious" extension. It's the extension 10 out of 10
mathematicians would pick if you asked them. In other words there's not
always a precise definition, but it's obvious.

Example: f:N->N given by f(n) = n, where N is the naturals. How would
you extend that "naturally" to the reals? Obviously you'd pick the
identity function on the reals; even though there are uncountably many
other extensions that take the same values on N.

Does that help?

A different situation: suppose that the set B has a structure (ordering,
group, topology) and A can inherit some or all of the structure in such a
way that a map defined on A has an extension which is, perhaps with some
added properties, unique.
Example: B can be an inner product space, A a dense linear subspace, and
f is a continuous linear map on A. Then f has exactly one extension which
is linear and continuous on all of B - the natural extension.
Or: B as before, A a closed subspace, f linear continuous on A, then
the "minimal" extension, defined as zero on the orthocomplement of A,
and extended by linearity, would be voted "natural".

Cheers, ZVK(Slavek).
.

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