Understanding Vector Spaces: Exploring Real Valued Functions on Arbitrary Sets

In summary: Hello Compuchip.Thankyou that was a great help. Using your example and a bit more thinking i think i am getting the idea.
  • #1
matheinste
1,068
0
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.
 
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  • #2
It is not clear what's causing you trouble here. Can you be more specific.

For instance, you want an example of what? :confused:
 
  • #3
Would this be a valid example?

Choose some set X. Let [itex]\{ x_1, x_2, \cdots, x_n \}[/itex] be a finite subset* of X. Let [itex]\{ y_1, y_2, \cdots, y_n \}[/itex] 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

Then define a function [itex]f: X \to \mathbb{R}[/itex] by
[tex]f(x) = \begin{cases} y_j & \text{if } x = x_j \text{ for some } j = 1, 2, \cdots, n \\ 0 & \text{otherwise} \end{cases}[/tex]
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).
 
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  • #4
Hello Compuchip.

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

Matheinste.
 
  • #5
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.
 
  • #6
matheinste said:
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 [itex]x \in X[/itex] 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)
 

What is a free vector space?

A free vector space is a mathematical concept that describes a set of vectors that can be added together and multiplied by a scalar without any additional constraints or restrictions.

How is a free vector space different from a regular vector space?

A free vector space is different from a regular vector space in that it does not have any predetermined basis or structure. In a regular vector space, the vectors are defined in terms of a specific basis, while in a free vector space, the basis can be chosen arbitrarily.

What are the properties of a free vector space?

The properties of a free vector space include closure under vector addition and scalar multiplication, associativity, commutativity, and distributivity. It also has a zero vector and inverse elements.

What is the significance of a free vector space?

A free vector space is significant in mathematics because it serves as a foundational concept for understanding more complex vector spaces, such as those found in linear algebra. It also has numerous applications in physics, engineering, and other scientific fields.

How is a free vector space used in real-world applications?

Free vector spaces are used in real-world applications to model physical quantities, such as forces, velocities, and electric fields. They also play a crucial role in computer graphics, where they are used to represent objects and their transformations in three-dimensional space.

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