TITLE: Intuition on an unexpected finite dimensional space
QUESTION [16 upvotes]: We had this exercise that uses Riesz' lemma to prove that a functional space actually has a finite dimension. I was curious to know what this space "looks" like, if we can find natural elements that would generate it. Here are the assumptions:

$E = \mathcal{C}^0([0,1], \mathbb{R}) $ is the space of continuous functions on $[0, 1]$, endowed with the norm $ ||f||_\infty = \sup\{|f(t)|, t \in [0, 1] \} $.
We have a closed linear subspace $F$ of $E$ such that $F \subset \mathcal{C}^1([0,1], \mathbb{R})$ (the space of continuously differentiable functions with continuous derivatives) and
$$ \exists C > 0, \forall f \in F, \; ||f'||_\infty \leq C \, ||f||_\infty $$

You can show that the unit ball of that space is compact for $ ||\cdot||_\infty$ (using Ascoli), hence by Riesz' lemma the space has a finite dimension (!)
If I follow the conclusion then I imagine that you could build up a finite family of functions that would generate the whole space. Is that true? What would these functions look like? Are they useful somewhere else? Are they known?

REPLY [9 votes]: I had this in the comments earlier, but  when I went to add an example, I found it too frustrating to edit in the comment box.
Suppose that $C = 1$. So the condition of interest on a subspace $F$ is: $\|f'\| \leq \|f\|$ for all $f \in F$.
Since $(e^x)'=e^x$  and $(e^{-x})' = -e^{-x}$ it is easy to see that the lines $L_1 = \mathbb{R} \cdot  e^x$ and $L_2 = \mathbb{R} \cdot e^{-x}$ both satisfy the condition. However, there cannot be a subspace $F$ satisfying the property which contains both $L_1$ and $L_2$. Such a space $F$ would need to contain $f = e^{x} - e^{-x}$, but one can check that $\|f'\| > \|f\|$, so this is impossible.
The conclusion from the above is that, for a given $C$, there is not just one example of a subspace $F$, but many. Moreover, there is not a largest one. That said, it might be interesting to ask if there is a finite bound on the dimensions of spaces satisfying the property for given $C$, and then search for examples realizing the maximal dimension.