High dimensional vector space references

Question

Is there any good text book or review papers that introduce high dimensional vector spaces and its peculiarities as compared to generic/low-dimensional vector spaces?

For example, high dimensional unit sphere $S^n$ ($n \gg 4$) has most of its mass near the boundary in an $n-1$ dimensional annulus. A treatise generically introducing linear algebra or vector spaces will not develop you intuition regarding the peculiarities of high-dimensional vector spaces, and will generally not spend a lot of ink on explicitly discussing the properties of them, since it is "generic".

Answer 1

It's possible you might be interested in Keith Ball's nice article ''An elementary introduction to modern convex geometry" and its references. It discusses many issues relevant to the geometric behavior you mention. It's available at http://page.math.tu-berlin.de/~Vybiral/GHS/ball.pdf

Answer 2

You really need to clarify your question because it doesn't make any sense as stated. I completely agree with Taladris-this is the kind of question that betrays the fact the student doesn't really understand the subject of thier question as well as they think.The theory of finite dimensional vector spaces i.e. basic linear algebra is completely general regardless of dimension as long as the dimension is finite. Another answerer wondered whether or not you meant infinite dimensional versus finite dimensional spaces. That was also my initial thought-but looking at the question again,I'm not sure if that's what you're asking about. It sounds more like you studied matrix algebra and "arrow" vectors in low dimensions.If so, you need to relearn linear algebra correctly now-that is,on abstract vector spaces using a good linear algebra book, such as Curtis. If you did mean infinite dimensional vector spaces,then you'll need to learn a considerable amount of hard analysis before you can tackle that mountain,from what it looks like to me.As long as the dimension is finite, the number of vectors in any basis for the vector space is unique.When you move into infinite dimensional spaces, that's no longer true. But this isn't even the most serious problem with infinite dimensional vector spaces. We can prove using the axiom of choice-or one of it's variants-that every vector space has a basis. The problem is that when dealing with infinite dimensional vector spaces, there's no clear way to construct a basis. Think about it-what does it mean to represent an element of a vector space as an infinite linear combination of linearly independent vectors? In some infinite dimensional spaces, such as those that occur in Fourier analysis, this idea can be made precise by convergent infinite series-but this is really an unusual case. In most cases, we can't even state exactly what this means, let alone come up with a procedure for constructing one. This is really what makes functional analysis and operator theory a completely different-and fascinating!-ball game then plain old linear algebra-you have to use other kinds of subsets of the vector space to specify the dimension. But to tackle it,you'll need at least a strong background in undergraduate real analysis on metric spaces in addition to a good understanding of abstract linear algebra.

Hope that helped clarify your question. I hope.

UPDATE Alright,your edited question makes much more sense now. But your question really isn't about vector spaces per se, it's about higher dimensional geometry-i.e. the topology and geometry of n-dimensional manifolds. In which case,what you really want is a recommendation for a book on modern differential geometry and topology, of which there are legion. The problem is I doubt from your question that you have the proper background yet to study this amazing topic. To even begin to study this topic,you need a good background in both basic topology and undergraduate real analysis,in addition to a strong understanding of linear algebra. Indeed, the questions you're asking actually require considerably more to even begin to answer,including a graduate course in algebraic topology. A good beginner's place to start is John McCleary's Geometry From A Differentiable Viewpoint, which not only presents the basic concepts of low dimensional differential geometry, but does so from a historical viewpoint the beginner will find very enlightening. It also contains many good recommendations for further study. That being said-you clearly have a lot to learn before you can begin to seriously study these topics,so I'd get started on that if I were you!

Answer 3

I suppose that by "higher dimensional" you mean spaces with dimensions higher than 2 or 3 or 4, as used commonly in physics, engineering, computer graphics, and high-school calculus and geometry.

If so, then look for books about "linear algebra". This is the study of vector spaces with arbitrary dimension (say $n$), including infinite dimensions. Most of the theory does not make any special mention of the special cases $n=2,3, \text{or }4$. A reasonable place to start is this Wikipedia page.

But, as you start to study more abstract vector spaces, don't forget what you know about the three-dimensional space in which we all live. 3D geometry and 3D vectors represented by "arrows" provide a valuable guide to intuition, even though many mathematicians would not view this as "real" linear algebra.