Which functions are dense in this Hölder-space? 
We are on the line $[0,T]$, let $a \in (0,1)$. Let $C^a$ consist of
  those real functions such that:
$|f|_a=\sup_{0\le s<t\le T }\{\frac{|f(t)-f(s)|}{(t-s)^a}\}<\infty$.
Denote the norm $\|f\|_a=|f|_a+|f|_\infty$. Where $|\cdot|_\infty$ is
  the standard supremum norm for continuous funtions. 
Using the mean-value theorem, we can show that the space
  $\mathcal{C}^1$, the space of continuously differentiable functions,
  is in $C^a$.

A theorem in a book I am reading states:

Let $f \in C^a$ with $a \in (0,1)$, and let $b\in(0,1)$ be such that
  $a+b>1$. The linear operator $T_f: \mathcal{C}^1\subset C^b\rightarrow
C^b$ defined as $Tf(g)=\int_0^\cdot f(u)g'(u)du$ is continuous with
  respect to the norm $\|\cdot\|_b$. By density, it extends (in a unique
  way) to an operator $T_f:C^b\rightarrow C^b.$

The author proves everything except from the density argument. From what I understand he refers to the classical result that if you have a dense subspace of a banach space, and you have a continous linear operator, you can extend it uniquely to the entire space? The problem I have is how is it dense. Because according to these links:
$C^1$ is not dense in Hölder space
Polynomials not dense in holder spaces
$\mathcal{C}^1$ is not dense in the Hölder space. I think in the links they only use the norm $|\cdot|_a$, not $\|\cdot\|_a$, but if it is not dense w.r.t to the first norm, it can't be with respect to the other norm that is a sum of the first norm and the supremum norm?
Do you see what I am missing? What kind of density argument is used here?
 A: The appeal to "density" is a little sloppy. I believe the following is what the author meant: 


*

*The closure of $\mathcal C^1$ with respect to $C^\beta$ norm is the little Hölder space $c^{\beta}$ (defined by the condition $|f(x)-f(y)| = o(|x-y|^\beta)$ as $x-y\to 0$)

*$C^{\beta+\epsilon}\subset c^\beta$ for every $\epsilon>0$. 

*Given $f\in C^\alpha$ and $\beta$ such that $\alpha+\beta>1$, pick $\gamma <\alpha$ such that $\gamma+\beta >1$. Then $f\in c^\gamma$, hence the density argument applies and shows that the integral $\int fg'$ is bounded by $C\|f\|_\gamma\|g\|_\beta$. The latter is controlled by $C\|f\|_\alpha\|g\|_\beta$, as desired. 


I don't have a suitable reference for (1), but this fact is not hard to prove directly: (a) smooth functions are in $c^\beta$; (b) $c^\beta$ is a closed subspace of $C^\beta$; (c) mollifying a $c^\beta$ function produces an approximating sequence that converges in $C^\beta$ norm; the key is that sufficiently small scales don't really contribute to the norm.
For reference, here is the relevant part of the source (page 25 in Selected Aspects of Fractional Brownian Motion by Ivan Nourdin). 

The reference Young [67] is to the classical article An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. (1936) 67, 251-282. Young actually works with larger spaces of functions of finite $p$-variation, and I couldn't locate a relevant density argument there.  A more accessible source is a pair of blog posts by Fabrice Baudoin: 


*

*Lecture 6. Continuous paths with bounded p-variation

*Lecture 7. Young’s integral
