The Meaning of SpaceDate: 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/ |
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