Mathematical symbols

Can anyone tell me what these symbols mean:

: (a colon on its own)
→

As some form of context, well I was reading about metric spaces and its in the definition:
http://en.wikipedia.org/wiki/Metric_space#Definition

and I've tried using the mathematical symbol wiki page but I don't get why M X M would "imply" the real numbers. I also can't find anywhere what the : symbol means in maths, bet its something silly . Is it just mean "same as?".

The notation $d: M \times M \to R$

means d is a mapping from MxM to R. That's the way people write mappings from sets to sets.


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Thanks . When they say MxM, is that MxM space?

It's the Cartesian product of two copies of M.


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What do you mean by cartesian product and what do you mean a copy?

What you need is a first course in functions. How to Think Like a Mathematician has a good explanation, IIRC, aside from being an excellent book.

I need to learn about Hilbert space, but all the definitions talk about metric space, linearised vector space and inner product so I'm learning about them first:s. Part of the reason I don't want to get bogged down is that every time I go onto wiki and try and learn something in maths, there is ALWAYS something in the definition I don't know and it basically goes round in a perpetual cycle of knowledge which I don't know. Regardless, I have no means of reading that book and I don't fancy reading a book entirely on maths :s

EDIT: I understand what cartesian product is now from wiki but I still don't understand how that applies to metric space (why would it not be two different sets, why does it explicitly have to be the same set??)

It sounds like you're trying to run before you know how to walk.


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Not really. I'm asking for help, its not like im taking a course or degree in mathematics... But your comment makes zero sense. Tell me why during A-Levels we use Euclidean space and NEVER learn about metric space? I'm just actually trying to learn things from scratch, if you can't or are unwilling to explain further, thats fine, but please don't assume or judge me like that.

I understand what cartesian product is now from wiki but I still don't understand how that applies to metric space (why would it not be two different sets, why does it explicitly have to be the same set??)

Not really. I'm asking for help, its not like im taking a course or degree in mathematics... But your comment makes zero sense. Tell me why during A-Levels we use Euclidean space and NEVER learn about metric space? I'm just actually trying to learn things from scratch, if you can't or are unwilling to explain further, thats fine, but please don't assume or judge me like that.

Because metric spaces are a much higher-level concept than that of Euclidean space. Euclidean space is pretty much the nicest kind of space - it admits a very simple norm. Metric spaces are much more general than normed spaces, and can be very peculiar. It requires correspondingly more knowledge to understand them. It's for the same reason that in primary school we learn about addition without *ever* learning complex numbers (which, after all, are simply the algebraic closure of the reals).

I *strongly* recommend reading books if you're going to become a mathematician without lecturers teaching you. There is almost certainly not going to be anyone here who is willing to put in the full day of tuition it might well take to get to metric spaces from A-level.

Why are you interested in Hilbert spaces, by the way?

The reason that the idea of "metric space" was dreamt up was to allow us to define a function that can tell you the distance between two points in a set, $M$. So you are always dealing with two points from the *same* set.

Mathematically we package these points up into a single object (an ordered pair) which the metric function associates with a distance i.e. a real number. The ordered pair belongs to the cartesian product of the set $M$, which is denoted by $M \times M$. $M \times M$ is, of course, itself a set (of ordered pairs).

For example the function $d(x,y) = |x-y|$ where $x,y$ are real numbers, is a metric on the cartesian product $\mathbb{R} \times \mathbb{R}$

Any given metric function must behave like the distances we know and love:

1. the distance between x and itself must be 0
2. the distance between x and y must be the distance between y and x
3. distances must obey the triangle inequality.

But the whole point is that you have build things up. You have to learn the basics first. You need at least a first course in linear algebra and one in analysis before you start on hilbert spaces. At this point you don't even know mapping notation which is ubiquitous.

Euclidean space is a metric space (under the Euclidean metric). The reason that metric spaces are not taught at a level is because they are not regarded as important enough to be on the syllabus I imagine.


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