Showing that a function is a Vectorspace-Isomorphism? Let $V$  be the real vector space $\mathbb{R}[X]$ and $M⊂\mathbb{R}$ with $d$ elements.
Furthermore let $U := \{  f \in \mathbb{R}[X]\ |\ \deg(f) ≤ d − 1 \} $ and $\Phi : V → \operatorname{Func}(M, \mathbb{R})$ which is a linear function given via $Φ(f)(m) := f(m)$.
How could I show step by step that $Φ|_U
: U → \operatorname{Func}(M, \mathbb{R})$ is a vector space isomorphism?
My idea is to show that: 
(1) It is a homomorphism (as it is a linear function, it is also a homomorphism, so that one is already proved, right?).
(2) It is injective.
(3) It is surjective.
Can you help me how is it possible to show those three circumstances?
(Can you please give me an explanation for every step, that I can understand everything?)
 A: As I noted in the comments, (1) is immediate from the comments, so I will focus on (2) and (3). Note that this only shows that $\Phi$ is a bijective vector space homomorphism, but every bijective vector space homomorphism is even a vector space isomorphism, because the inverse of any linear function is again linear (the analogous statement is not true, e.g., for continuous functions).
Also I should note that I am being lazy and, unlike what you asked for, I am not really showing every step in what follows, just giving pointers so that you can find resources which will explain the answers to you.
(2) We want to show that, $\Phi(f) = \Phi(g) \implies f=g$. This is basically the statement that any polynomial of degree less than or equal to $(d-1)$ is determined uniquely by its values at $d$ points -- this can be shown, for example, using Lagrange interpolation.
(3) We want to show that any function from $M$ to $\mathbb{R}$ can be described by the action of a polynomial. In other words, given data points $(x_1, y_1), \dots, (x_d,y_d)$, we want to find a polynomial such that $f(x_i)=y_i$ for all $i=1,\dots,d$. That this is possible is again discussed and proved in the theory of Lagrange interpolation.
Since there is such an extensive literature out there discussing Lagrange interpolation and other polynomial interpolation methods (e.g. Newton interpolation), I figured it would be best if I don't try to reinvent the wheel and explain all of it again here myself. If you have followup questions please of course let me know.
