I'm a student in the French equivalent of college, and I had a question about something I saw in an old competitive exam. It was stated : "Let $F$ be the $\mathbb{C}$-vector space composed of the functions $f : \begin{array}{rcl} \mathbb{R} & \longrightarrow & \mathbb{C} \\ x & \longmapsto & x^k\rho^x e^{i\theta x}\end{array}\ \ \ $ where $k \in \{0, 1, 2\}$, $\rho \in ]0, +\infty[$, $\theta \in ]0, 2\pi ]$."
I suppose it is meant to be a vector space for $+$ and not for $\times$. Nevertheless, the exam paper admitted it was a vector space, without demonstrating it, and without even specifying whether it was additive or multiplicative. But I have some doubts about it : $F$ does not even seem to contain the identity element for $+$, which would be the function $\tilde 0$. It looks like it could be a vector space for the operator $\times$ though...
Does anyone agree ? Could someone explain this to me ? Maybe I'm completely wrong but I'm not sure about it either...
Answer : the exam paper actually specified that $F$ is the vector space $\textit{spanned}$ by those functions. My bad !