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.
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.