Proof of vector subspace on R functions Let $F (\Bbb R,\Bbb R)$ be the vector space of all the functions from $\Bbb R$ to $\Bbb R$. For what values ​​of $k\in\Bbb R$, the set 
$W = \{f\in F (\Bbb R,\Bbb R)\mid f (1) = k\}\leqslant F$?

Any help?
As you say, the conditions for $W$ to be a linear subspace are that it should be closed under arbitrary linear combinations. That is for $f,g\in W$ and any $\alpha, \beta$

$$
\alpha f+ \beta g \in W
$$

In particular this should be true for $\alpha=\beta =0$ meaning that the null function should also be in $W$. 

Can you take it from here?
If $W $ is a subspace, then the null-function belongs to  $W $

Can you proceed?