I'm currently trying to explain divided differences with a view to defining the Newton form of the interpolating polynomial. I'm using the definition:
The $k$th divided difference of a function $f$ at the sites $t_{i}, ..., t_{i+k}$ to be the leading coefficient (i.e. the coefficient of $x^{k}$) of the $k+1$st order polynomial that agrees with $f$ at $\textbf{t}=(t_{i}, ... ,t_{i+k})$. We denote it by $$[t_{i}, ..., t_{i+k}]f.$$
And I wanted to mention that it is linear in $f$. I was hoping to say something along the lines of:
The function $[t_{i}, ..., t_{i+k}]f: V \rightarrow \mathbb{R}$ is linear in $f$ in that if $f = \lambda g + \mu h$ then $$[t_{i}, ..., t_{i+k}]f=\lambda [t_{i}, ..., t_{i+k}]g + \mu [t_{i}, ..., t_{i+k}]h.$$
But I realized that I'm not exactly sure what the space $V$ actually is. Is it the space of all real-valued functions?