Can anybody help me to interpret the following, if $V$ is the real vectorspace $\Bbb R[X]$ and $M$ is subset of $\Bbb R$?

Question

So, as I said $V$ is the real Vectorspace $\Bbb R[X]$ and $M\subset \Bbb R$.

Furtermore, $Φ : V → Func(M,\Bbb R)$ which is a linear function given via $Φ(f)(m) := f(m)$.

What I do not really understand how the function $Φ$ works and what is $Func(M,\Bbb R)$ and what does $ Φ(f)(m):=f(m)$ mean.

Thanks for your help in advance.

Answer 1

$Func(N, \Bbb R)$ is the set of ALL functions from $M$ to the real numbers.

$V$, on the other hand, is the set of polynomials in $X$ over the reals (which happens to be a vector space).

Let's look at an example: $M = \{2, 3\}$.

Now a typical element of $V$ is something like $p(x) = x^2 - 1$. What's $$ \Phi(p)? $$ Well, it's supposed to be a function on $M$. So you have to say what the value of $\Phi(p)$ is for each element of $m$. The rule tells you: $$ \Phi(p)(2) $$ for instance, is the value of the polynomial $p$ at $2$, i.e., $p(2) = 2^2 - 1 = 3$. Similarly, $\Phi(p)(3)$ is $p(3)$, which is $3^2 - 1 = 8$.