The Meaning of Space

Date: 07/23/97 at 14:20:15
From: Anonymous
Subject: I have a question on Space

Dr. Math,

I am neither a student nor a teacher, but I am interested in math. 
I try to read advanced math for laymen, but there are not many books 
out there. Anyway, I have a question for you.

I have seen this term 'Space' mentioned a lot. Is that a mathematical 
object? I have seen it in "Vector Space," "Banach Space," "Hilbert 
Space," but it seems that they are not the same thing. 

If I remember right, concepts like a "Ring" or "Field" have unique 
definitions. I wonder whether that is true for "Space."

Thank you so much.

Peter

Date: 07/23/97 at 19:08:25
From: Doctor Wilkinson
Subject: Re: I have a question on Space

The term "space" by itself does not have a unique definition, as you 
have correctly noticed.  As it happens, the three examples you give 
are all vector spaces, "vector space" being the most general term, and 
"Hilbert space" the most specialized. 

What these various kinds of space have in common is that they 
generalize ordinary ideas of geometry.  Thus ordinary three-
dimensional space can be thought of as a vector space. It also
happens to be a simple example of both a Banach space and a Hilbert 
space. Banach spaces and Hilbert spaces only get really interesting, 
however, when the number of dimensions becomes infinite.

Usually the term indicates some kind of abstraction from ordinary 
geometry. The most general terms are "vector space," which roughly 
speaking is concerned with algebraic properties; and "topological 
space," which is concerned with properties of "closeness," things more 
related to calculus, for example.

Some books on mathematics that you might enjoy are _Mathematics and 
the Imagination_ by Kasner and Newman, _What is Mathematics?_ by 
Courant and Robbins, and _Journey through Genius_ by William Dunham.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/