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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?

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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.

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  • $\begingroup$ Ok for the second point. For the first point that means that the subspace has dimension two? Without solving the diff. eq. how can you conclude that you just need two functions for the basis and not for example three functions? $\endgroup$ – Ubaldo Tiberi Nov 4 '15 at 12:53

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