In differential geometry, tangent vectors are considered operators.
At point p, the local tangent space is defined as 
$$
T\_p(M)=\{X^i\partial\_i|X\in R^n\}
$$
This is quite easy to understand for me.


However, I study information geometry recently and get stuck with the tangent vectors defined on statistical manifold.


As <http://en.wikipedia.org/wiki/Information_geometry> points out, the tangent vectors defined at point $p\_\xi$ are $\partial\_ip\_\xi$ in mixture representation. I really cannot understand why tangent spaces can be defined like this! It does not make any sense to me. And I cannot digest the explanations on wikipedia. 


Can anyone help me understand it？
Thanks in advance！


I can understand the previous problem now. 
Is there anyone who has studied information geometry before.
I have a new question.


on
<http://en.wikipedia.org/wiki/Information_geometry>
$D[\partial\_i\partial\_j||\cdot]= D[\cdot||\partial\_i\partial\_j]=-D[\partial\_i||\partial\_j]$.
I think they should all equal to 0.
Here is my reason:


because $D[\partial\_j||\cdot]=0$,
we have $$
0=\partial\_iD[\partial\_j||\cdot]=\partial\_iD((\partial\_j)\_p||p)=\partial\_i\partial\_jD(p||p)=D((\partial\_i\partial\_j)\_p||p)=D[\partial\_i\partial\_j||\cdot]
$$


But the true result seems to support such equation:
$$
\partial\_iD((\partial\_j)\_p||p)=D((\partial\_i\partial\_j)\_p||p)+D((\partial\_i)\_p||(\partial\_j)\_p)
$$


Why?



Why tangent vector of statistical manifold is a function?
There are situations where you think of a function space as an infinite-dimensional manifold. Then the tangent space is also infinite-dimensional, and also naturally a function space. The tangent space to a product manifold is the direct sum of the tangent spaces, and a function space is like a (contiuum) product, so its tangent space should be a product over the same index set, i.e. consist of functions on the same domain.


