I found out what the Hausdorff-Besicovich Dimension was (from a Gleick book), and what Topological Dimensions is all about, but Banach Space, Hilbert Space, and other spaces like them I can’t fathom. What are they? I know I already exist in the third Hausdorff-Besicovich Dimension, which is the same as the Third Topological Dimension, but can I enter into the mysterium of Banach Space, for example?
You’re kidding right? This is the kind of stuff you pay college professors thousands and thousands of dollars to learn about. I remember Hilbert space had something to do with infinite dimensions with certain special properties. Never heard of Banach spaces though…
Maybe if you paid me I’d begin to recall some more details 
Spaces in mathematics have nothing to do with the physical space we live in. You can’t go into them, any more than you could catch fish on the Number Line. They are abstract places where abstract mathematical objects live.
A Hilbert space is an infinite-dimensional, complete metric space with a scalar product defined.
A Banach space is the same as a Hilbert space except it only needs a norm, not a scalar product.
“Complete” means that every Cauchy sequence has a strong limit in the space
A Cauchy sequence is…ah, the hell with it.
I have nothing of value to add. I would just like to comment on how happy I am to see a don willard thread that uses lower case characters in the subject line.
I’ll try and give you an intuitive idea of what Banach and Hilbert spaces are.
First, there’s what’s called a vector space. The definition of a vector space is a long list of algebraic properties, if I listed them here I doubt it would help your intuition any, but I imagine you’ve heard of vectors (magnitude and direction) before, that’s a helpful model to keep in mind. Vectors in a vector space are not quite that specific however–“vector” in this sense is basically left undefined, they are only required to satisfy certain algebraic properties regarding addition of vectors and scalar multiplication–vector space is just an algebraic structure.
Then there’s a normed vector space–the norm basically means that you can now talk about the length of the vectors. As a result, you also get a metric defined on the vector space, so that it’s now also a metric space, meaning you can now talk about the distance between two vectors. So now we’ve put a kind of “spatial” structure on the vector space.
A complete metric space is one where Cauchy sequences always converge. I’ve always kind of thought of this as meaning there are no “holes” in the metric space. Kind of a “least upper bound” property, if you’re familiar with that. For example, if we’re talking about the space of rational numbers, consider the sequence of numbers {3, 3.1, 3.14, 3.141, 3.1415,…} (digits of pi). This is an example of a Cauchy sequence, but this sequence doesn’t converge because pi is not a rational number–there’s basically a “hole” there in the rational numbers where this sequence is trying to converge, but can’t.
Anyway, so a Banach space is a complete normed vector space.
Taking it one step further, given a vector space you can also have an inner product on a vector space. An intuitive way of thinking of this is that now we can talk about angles between vectors, so we’ve actually got a geometrical structure now. An inner product space is a normed space, but a normed space is not necessarily an inner product space.
A Hilbert space is a complete inner product space.
I know Hilbert spaces are used quite a bit in physics (quantum mechanics, for example), but you’ll have to get a physicist in here to help you much with that.
I used Hilbert spaces frequently as an electrical engineer in grad school. Never dealt with Banach spaces (outside of real analysis classes :))
Excellent explanations, btw.
Arjuna34
The term “space” is probably the deceptive bit, since you’re trying to tie it to the common dictionary definition of space. It’s not being used that way, it’s being used in the sense of other mathematical concepts, like a group (being a bunch of numbers with a single operation such as addition), or a field (being a bunch of numbers with two operations such as addition and multiplication that interact). A mathematical field has nothing to do with a field full of cows and horses.
Similarly, a space is sort of the next step up, it’s a bunch of numbers AND directions. Thus, as someone mentioned, the Real number line is both a field and a 1-dimensional space. The Complex plane is both a field and a 2-dimensional space.
These are all used to create mathematical models for various purposes (including trying to create a model of real space, but there are other models as well.) Different properties are required for different modelling purposes, and thus you get metric spaces, complete spaces, etc as neatly outlined by Cabbage above.