I'm having some trouble with this problem that I have been working on for the past couple of hours. I was wondering if someone could please point me in the right direction.

Consider the set $S = \{A, B, C, D\}$. I need to find a basis for the vector space $F(S, \mathbb{R})$ (the set of functions that map from $S$ to $\mathbb{R}$). Then, I need help in defining an inner product on the vector space to make the basis orthonormal?

My attempt:

I believe the basis should be a set of functions, for example, $S = \{f(A) = 1, f(B) = 2, f(C) = 3, f(D) = 4\}$. I do not have much reasoning for this, though. I am just using past problems to guide me.

Any assistance is appreciated.


The first thing to notice is that we can map $F$ to $\mathbb{R}^4$ by mapping each $f \in F$ to its tuple of values at the "points" $A$, $B$, $C$ and $D$ i.e.

$\rho : F \rightarrow \mathbb{R}^4 : \rho(f) = (f(A), f(B), f(C), f(D))$

Note that $\rho$ is a 1-1 mapping so we can define its inverse:

$\rho^{-1} : \mathbb{R}^4 \rightarrow F : \rho^{-1}( (a,b,c,d) )= f : f(A)=a, f(B)=b, f(C)=c, f(D)=d$

We can then turn $F$ into a vector space by "pulling back" the natural vector space structure on $\mathbb{R}^4$:

$f + g = \rho^{-1}(\rho(f) + \rho(g)) = \rho^{-1}((f(A)+g(A), f(B)+g(B), f(C)+g(C), f(D)+g(D)))$

$kg = \rho^{-1}(k\rho(f)) = \rho^{-1}((kf(A), kf(B), kf(C), kf(D))) \text{ for } k \in \mathbb{R}$

and we can "pull back" the standard definition of inner product as well:

$<f,g> = \rho^{-1}(<\rho(f),\rho(g)>) = \rho^{-1}((f(A)g(A), f(B)g(B), f(C)g(C), f(D)g(D)))$

The inverse mapping $\rho^{-1}$ can then be used to map any orthonormal basis in $\mathbb{R^4}$ to an equivalent orthonormal basis in $F$. As Berci points out in their answer, a natural choice is the image of the basis $\{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)\}$ - but that is not the only choice available.


What you provided is (the set of values of) a single function $S\to\Bbb R$ and not a set of functions.

For $X\in S$, let $e_X:S\to\Bbb R$ assign $1$ to $X$ and $0$ to every other elements.
Verify that $\{e_A, e_B, e_C, e_D\}$ is a basis.

Then define the inner product the usual way on coordinates w.r.t. this basis.


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