A total set in a NLS is one whose linear span is dense in the set. e.g. $A = \{1,x, x^2,...\}$ is total in $(C[a,b],\Vert\cdot\Vert_{\infty})$
I find it easier to talk about total sets than dense sets (due to incompetence of LaTeX) so I will do so for the remainder of this post. (side note: if someone can tell me what the code is for the proper use of the summation sign (with stuff on the top and bottom) in LaTeX, I would be very grateful).
Edit: I only know the definition of density which uses convergence, which requires a norm. i assume that in all areas of maths, density requires a norm or a m. space etc.
So anyway, back to maths. From now on when we talk about $C[a,b]$, assume a and b are any two real numbers already given to us (obviously). We are in $C[a,b]$, or $C(-{\infty}, {\infty})$.
The short version: Is there a "relatively simple" way to tell if a set is total on 1) ([a,b],arbitrary norm) , and 2) on (R, arbitrary norm)?
Answer attempt to 1) After finishing writing the long version below, I have just found (in a book) the Stone-Weierstrass Theorem. This helps things. But for any non-compact set, is it true that there are NO dense sets, no matter what the norm is? That would settle a few more things 9but not completely answer the questions)
The long version:
Here is my first question:
Define $\Vert\cdot\Vert_{i} = (\int_{a}^{b}\parallel \cdot\parallel^{i})^{\frac{1}{i}}$.
The sup norm $\Vert\cdot\Vert_{\infty}$ is the "strongest norm" out of {$\Vert\cdot\Vert_{\infty}$ ,...$\Vert\cdot\Vert_{2}$ ,$\Vert\cdot\Vert_{1}$} because convergence in $\Vert\cdot\Vert_{\infty}$ implies convergence in... implies convergence in $\Vert\cdot\Vert_{2}$ implies convergence in $\Vert\cdot\Vert_{1}$. (Although the first step doesn't actually make sense, what I am really saying is that $\Vert\cdot\Vert_{\infty}$ converges to $\Vert\cdot\Vert_{i}$ for every i, and I am including it conveniently in the statement). I think that the statement is right anyway, but I am not sure how to prove it (first question).
My second question is: Is A total in C($-{\infty}$, ${\infty}$) with the sup norm i.e. $\Vert\cdot\Vert_{\infty}$ ? Hopefully you can see how the two questions are related. If it were true that A is total in $(C(-{\infty},{\infty}),\Vert\cdot\Vert_{\infty})$, then A would be total in all the other norms, assuming my intuition about the other statement is correct.
Further question: Is B = {1,1/x, 1/x^2,...} total in $(C[a,b],\Vert\cdot\Vert_{\infty})$ ? If not, is it total in any other sets (e.g. function converging to 0)
Further further question: I know that {sin(pi*x), sin(2*pi*x),...} is total w.r.t. $(C[a,b],\Vert\cdot\Vert_{2})$ (but not total in [a,b] w.r.t. not the sup norm). What about in $(C(-{\infty}, {\infty}),\Vert\cdot\Vert_{2})$ ?