dg.differential geometry - How to shown that the Tangent Bundle of a vector space is a Vector Bundle

Hello,

I have the following question about the tangent bundle $T_M =
bigcup_{p in M} {p} times T_p M$ defined on a manifold $M$ of class $C^r$
modeled on a normed space $X$. My problem is showing that the tangent bundle
also forms a vector bundle. I found the following definition of a vector
bundle

A vector bundle is a tuple $E, B, pi, F, mathcal{T}$ where $E, B$ are
topological spaces, $pi : E rightarrow B$ a continuous surjection, $F$ a
normed metric space, $mathcal{T}$ is a family ${U_i, varphi_i }_{i in I}$
of homeomorphism $varphi_i : U_i times F rightarrow pi^{- 1} (U_i)$ with
$B = bigcup_{i in I} U_i$ such that

• $forall b in B succ pi^{- 1} ({b})$ has the structure of a
  normed vectorspace




• $forall i in I$ we have $forall x in U_i$ and $forall v in F$
  that $pi (varphi_i (x, v)) = x$




• $forall i in I, x in U_i$ the map $varphi_i^{(x)} : F rightarrow
  pi^{- 1} ({x})$ defined by $varphi_i^{(x)} (v) = varphi_i (x, v)$ is a
  linear function between the vector spaces $F$ and $pi^{- 1} ({x})$

We call

• $E$ the total space of the vector bundle




• $B$ the base space of the vector bundle




• $pi$ is the projection map of the bundle




• $mathcal{T}$ is called a trivialization and $(U_i, varphi_i)$ is
  called a trivializing neighborhood.
  end{itemize}

Now for the tangent bundle it is easy to see that $T_M$ is the total space and
$pi : T_M rightarrow M : (x, v) rightarrow x$ is the projection, $M$ is the
base space and I think we can equate $F$ with $X$, but how do you go on in
finding a trivialization. I thought first about using the induced atlas on
$T_M$ (that makes the tangent bundle a differentiable manifold of class $C^{r
- 1}$ modelled on $X times X$ but its mappings has not the correct format.

My problem with using the induced atlas as a trivialization is that it is of the form ${U_i, varphi_i }_{i in I}$ $varphi_i : pi^{- 1} (U_i) rightarrow varphi (U_i) times X$ and using
$varphi_i^{- 1} : varphi (U_i) times X rightarrow pi^{- 1} (U_i)$ I'm
almost there but I have still not found a homeomorphism of the form $U_i
times X rightarrow pi^{- 1} (U_i)$. The book I'm reading is talking about a
tangent space and says it is vector bundle but does not define a vector
bundle at all, so I looke up the definition of a vector bundle and failed to

Maybe I'm missing on the definition of a vector bundle (most examples I found
on the internet are about finite dimensional spaces ).

Can anybody help me?

Thanks a lot in advance

Marc Mertens