Dense subspaces How does one go about proving the following statements?
(a) $\operatorname{Lip}[a,b]$ functions are dense in absolutely continuous functions on $[a,b]$ under the variation norm - (Another doubt: what is the variation norm?)
(b) The subspace of continuous, piecewise linear functions is dense in absolutely continuous functions on $[a,b]$ - (A direct proof of this is known). Prove the statement using the fact the step functions on $[a,b]$ is dense in $\mathcal{L}^1[a,b]$.
Any help is appreciated.
 A: The general strategy for proving that a subspace $S$ is dense in a normed vector space $(X, \| \cdot \|)$ is as follows: for every $\epsilon > 0$ arbitrarily small, and every $x \in X$, show that there is an element $s \in S$ such that $\|x - s\| < \epsilon$. In other words, show that every vector in the bigger space is arbitrarily close to something contained in the subspace. If you can do this, then $S$ is dense in $X$.
For example, to approach (b), you would start by taking any absolutely continuous function $[a,b]$, and then seeing if, for any small $\epsilon > 0$, you can find a piecewise linear function $g$ that is very, very similar to $f$, so that $\|f - g\| < \epsilon$.
The variation norm is the norm
$$ \| f\|_{BV} = \sup_{P} \sum_{i} | f(x_{i+1}) - f(x_i) |, $$
where the supremum is taken over all partitions $P = (a = x_0, x_1, \ldots, x_n = b)$ of all sizes. You can think of this as adding up all the changes in $f$ over the interval, regardless of whether they are increases or decreases.
