TITLE: Does there exist a 3-dimensional subspace of real functions consisting only of monotone functions?
QUESTION [13 upvotes]: This is Exercise 1.O from the book 
Van Rooij, Schikhof: A Second Course on Real Functions.

The set of the monotone functions on $[0,1]$ contains all polynomial
  functions of degree $\le 1$. These form a two-dimensional vector space. Does the set of
  all monotone functions contain a three-dimensional vector space?

REPLY [7 votes]: If $V$ is a vector space of monotone functions and $f_1, f_2, f_3 \in V$ then $\left(f_i(0), f_i(1)\right) \in \mathbb{R^2}$ are three vectors in a two-dimensional space and therefore dependent. That means that there is a non-trivial linear combination of $f_1, f_2, f_3$ that vanishes at $0$ and $1$ and because it is also monotone, it is identically zero. So any three elements of $V$ are linearly dependent.