# Bases for two linear subspaces in the sets of all the functions $f:\mathbb{R} \to \mathbb{R}$ and $f:\mathbb{R}^n \to \mathbb{R}^n$, respectively.

Let $X$ the set of the all the functions $f:\mathbb{R} \to \mathbb{R}$. Let $Y:=\{f\in X : \frac{d^2 f}{d x^2}+f=0\}$. Is that a linear subspace (I would say yes)? How to write a base for $Y$?

Furthermore, let $A$ the set of all the functions $f:\mathbb{R}^n \to \mathbb{R}^n$. Let $B:\{f\in A: ||\frac{\partial f}{\partial x}||\le 1, \forall x \in \mathbb{R}^n \}$. Is that a subspace (I am attempted to say yes)? How to write a base for such subspace?

Regarding your first question, the space $Y$ is the space of all solutions to the linear differential equation $f'' + f = 0$ or $f'' = -f$. This is a second order linear differential equation which, if you will solve using the regular techniques, will yield two linearly independent solutions which will form a basis for $Y$. Since this equation is so simple, you don't even need to really solve it (can you think of two linearly independent functions $f$ such that their second derivative $f''$ is $-f$?).
Regarding your second question, $B$ won't be a subspace. To see this clearly, take $n = 1$ and ask yourself whether $B$ is closed under scalar multiplication or addition.