What Are Real Valued Functions on Arbitrary Sets in Vector Spaces?

[matheinste]

Hello all.

While looking at vector spaces leading up to multilinear mappings i am having trouble right at the start with the idea of the set of all real valued functions on an arbitrary set which vanish at all but a finite number of points. The author ( Wasserman, Tensors and Manifolds ) does not explain much about them i suppose because he thinks the definition self evident.

Is there any restriction on what sort of objects these sets can contain ( i suppose arbitrary means there is no restriction ) or any restriction on the types of functions other than those in the definition.

An example would be helpful.

Matheinste.

 

[quasar987]

It is not clear what's causing you trouble here. Can you be more specific.

For instance, you want an example of what?

 

[CompuChip]

Would this be a valid example?

Choose some set X. Let $\{ x_1, x_2, \cdots, x_n \}$ be a finite subset* of X. Let $\{ y_1, y_2, \cdots, y_n \}$ be a set* of non-zero real numbers
*) actually, some of the elements may be the same, so if you want to be precise, you should probably make it a sequence. Anyway, you know what I mean[/size]

Then define a function $f: X \to \mathbb{R}$ by
$$f(x) = \begin{cases} y_j & \text{if } x = x_j \text{ for some } j = 1, 2, \cdots, n \\ 0 & \text{otherwise} \end{cases}$$
and you have your function (in fact, this lists them all). Note that the function is nowhere near continuous, injective or surjective. But it is definitely a function (as in: a mapping from one set to another, or a relation on the Cartesian product).

 

[matheinste]

Hello Compuchip.

Thankyou that was a great help. Using your example and a bit more thinking i think i am getting the idea.

Matheinste.

 

[matheinste]

Hello CompuChip.

Regarding your reply in post #3.

Two questions:-

1:- Am i correct in saying that the construction in your example allows us, for whatever 'objects' are in X, to assign any value, depending on the set Y. In other words for any x in X we can assign the corresponding, in this case real number, object in Y.

2:- If so i can't grasp in what sense this lists all real valued functions on the set X but i am sure it will become obvious with a pointer in the right direction.

Thanks for your help so far. Any more help from anyone would be much appreciated.

Matheinste.

 

[CompuChip]

1:- Am i correct in saying that the construction in your example allows us, for whatever 'objects' are in X, to assign any value, depending on the set Y. In other words for any x in X we can assign the corresponding, in this case real number, object in Y.

2:- If so i can't grasp in what sense this lists all real valued functions on the set X but i am sure it will become obvious with a pointer in the right direction.

Actually, that's just how we usually define a function. For example, let X and Y both be the set of real numbers. Then I can specify a function f: X -> Y by saying what the function value in Y is for each value of X. For example, I can say: f is the function which maps any number x in X to the number x², which is usually just written f(x) = x².
But of course, X and Y can be any sets. Now if I specify for each $x \in X$ what the value f(x) is, I have defined a function. If X is finite we can do that by just listing them all, otherwise we have to find a more convenient way (like the f(x) = x² notation). Or you can combine the notations and say something like

f(1/2) = 3
f(12,4345) = 19
f(x) = x if x is an integer
f(x) = 0 if f(x) isn't fixed by the rules above (i.e.: for all other x)

which would also define a function from R to R (or actually, from any set containing {1/2, 12.4345} and all the integers to any set containing all the integers)