Is the vector space finite-dimensional? > The vector subspace $U$ of $\mathbb{R}[X]$ (polynomials) > > $U:=${$f∈\mathbb{R}[X] | f(\alpha+1)-f(\alpha)=f(\beta+1)-f(\beta) \alpha,\beta∈\mathbb{R}$} > > is finite-dimensional? I know that $\mathbb{R}[X]$ is finite-dimensional, because it has a basis {$1,x,x^2,...,x^n$} and its dimension is $\dim(\mathbb{R}[X])=n+1$. So theoretically a subspace of $\mathbb{R}[X]$ must be finite too. Is that right? Otherways I don't know how to demonstrate that. Any help please

No, $\mathbb R[X]$ is infinite-dimensional. A basis is $\\{1, x, x^2, \ldots\\}$ (there is no $n$ where this stops). But $U$ is indeed finite-dimensional.

Hint: if $f$ is a polynomial of degree $n$, what is the degree of $f(X+1)-f(X)$?