Reference request: Topological space of polygonal chains and its properties I'm interested in approximations of $C^1([a,b])$-functions by polygonal chains:

image File:NURBstatic.svg by User:WulfTheSaxon licensed under GNU Free Documentation License
A polygonal chain can be defined by a finite sequence of pairs $(x_i, y_i)$ with $a = x_0 < x_1 < \ldots < x_n = b$ (i.e. the sequence of the corners). Thus I am only interested in polygonal chains which can be represented by continuous functions $[a,b]\rightarrow \mathbb R$ whose graph are composed of line segments.
Lets take the later description and let $P([a,b])$ be all polygonal chains $[a,b] \rightarrow \mathbb R$. I guess that $P([a,b])$ is a vector space with pointwise addition and scalar multiplication.
My Question: Are there papers or books which discuss possible topological structures on $P([a,b])$ and its proberties? Is there a mathematical field which investigate those structures?
For example one can equip $P([a,b])$ with the supremum norm $\|\cdot\|_\infty$ so that it become a normed vector space...
Reason for my question: Let $s_p(x)$ be the current slope of the polygonal chain $p \in P([a,b])$ at the position $x\in [a,b]$. For any $p\in P([a,b])$ one can define
$$\|p\|_1 = \sup \{|p(x)|: x \in [a,b]\} + \sup \{|s_p(x)|: x \in [a,b]\}$$
I have the feeling that $\|\cdot\|_1$ is a norm for $P([a,b])$. I also have some hypotheses about the properties of $(P([a,b]),  \|\cdot\|_1)$:


*

*The completion of $(P([a,b]),  \|\cdot\|_1)$ is isomorph to a subset of $C([a,b])$, i.e. convergence in $\|\cdot\|_1$ means convergence to a continuous function.

*Any bounded and closed subset of the completion of $(P([a,b]),  \|\cdot\|_1)$ is compact.

*...


Now I can try to prove or disprove my hypotheses by myself. But I do not want to reinvent the wheel. That's why I am interested in works where the above hypotheses are already discussed. Thanks in advance for answering my question.
 A: The more standard term for your "polygonal chain" is "piecewise linear function".  Most of the interest in functional analysis is in complete function spaces; the piecewise linear functions are not complete in any common metric that I know of, so for the most part they will not be considered as a space by themselves, but rather as a subspace of a larger space.
Your norm $\|\cdot\|_1$ is the $W^{1,\infty}$ Sobolev norm.  If you complete the piecewise linear functions under this norm, you will get the Sobolev space $W^{1, \infty}([a,b])$ consisting of all absolutely continuous functions with essentially bounded derivative.
It will not be the case that all closed bounded sets are compact.  This is false in every infinite-dimensional normed space by Riesz's lemma.
It will be true that a sequence bounded in your $\|\cdot\|_1$ Sobolev norm has a subsequence converging uniformly to a continuous function (which is not necessarily piecewise linear), but the convergence need not hold in $\|\cdot\|_1$ norm.  This follows from the Arzela-Ascoli theorem since a set bounded in $\|\cdot\|_1$-norm is equicontinuous, and is a simple example of a Sobolev embedding theorem: the inclusion map from $(P([a,b]), \|\cdot\|_1)$ (or its completion) into $(C([a,b]), \|\cdot\|_\infty)$ is a compact operator.
