Variation of a function What exactly is the variation of a function ? Is it a distace or an element of some space
The total of a real valued function $f\colon [0,t] \mapsto \Re $ is as below say $\pi = \{0=t_0,t_1,\cdots , t_n=t\} $ then the $p^{th}$ variation of $f$ is 
$$ \lim_{\Vert \pi\Vert \to 0}V^p(f)=\sum_{i=1}^n \vert f(t_i)-f(t_{i-1})\vert^p $$
Now say we have $g\colon [0,t] \mapsto \mathcal{X}$ where $\mathcal{x}$ is some linear, normed,Complete space. it seems natural to say variation of $g$ over $[0,t]$ is 
$$ Z_n=\sum_{i=1}^n \Vert g(t_i)-g(t_{i-1})\Vert^p $$ where 
$\Vert.\Vert$ is the norm of $\mathcal{X}$
If it exists what is the Total variation in this case is it $\lim_{n \to \infty}Z$ ie a value in $[0,\infty]$ or is it some $x$ such that $\Vert Z_n -  x \Vert \to 0 $
say for wiener process $W(s)$ it quadratic  variation over  $[0,t]$ is $t$ does it mean 
$$ \lim_{\Vert \pi\Vert \to 0}V^2(W)=\sum_{i=1}^n \vert W(t_i)-W(t_{i-1})\vert^2 $$ in $L^2(\Omega) $ norm or does it mean some $X \in L^2(\Omega)$ 
 A: Consider the case where $W$ is a standard Brownian motion. Given a partition $\pi  = \{ 0 = t_0  < t_1  <  \cdots  < t_n  = t\}$ of the interval $[0,t]$, define
$$
[W]^\pi   = \sum\limits_{i = 1}^n {[W(t_i ) - W(t_{i - 1} )]^2 }. 
$$
Let $|\pi | = \max _i (t_i  - t_{i - 1} )$ denote the mesh of the partition. As is well known, given a sequence of partitions $\pi_n$  with $|\pi_n| \to 0$ as $n \to \infty$, $[W]^{\pi _n }  \to t$ in the $L^2$ norm, that is ${\rm E}\big[([W]^{\pi _n }  - t)^2 \big] \to 0$. Hence, in particular, $[W]^{\pi _n }  \to t$ in probability, and thus, by definition, the quadratic variation of $W$ is given by $[W]_t = t$. The key point here is that the limit $t$ in the right-hand side of $[W]^{\pi _n }  \to t$ plays a role of a random variable, rather than of a constant. Of course, in this particular example, $t$ is also a constant, but in general the limit is a random variable, which is defined using convergence in probability. 
EDIT: A much simpler yet instructive example is given by the Poisson process $N$. It is readily seen that the variation of $N$ is given by
$$
\mathop {\lim }\limits_{|\pi _n | \to 0} \sum\limits_{i = 1}^n {|N(t_{i} ) - N(t_{i-1} )|} = N(t),
$$
and the quadratic variation of $N$ is given by
$$
\mathop {\lim }\limits_{|\pi _n | \to 0} \sum\limits_{i = 1}^n {[N(t_{i} ) - N(t_{i-1} )]^2} = N(t).
$$
You may wish to consider different modes of convergence, as an exercise.
