I Prove $$T_{p}M$$ is a vector space with the axioms

[Delong66]

Suppose M is a manifold and $$T_{p}M$$ is the tangent space at a point $$p \in M$$. How do i prove that it is indeed a vector space using the axioms:
Suppose that u,v, w $$\in V$$. where u,v, w are vectors and $$\V$$ is a vector space

$$u + v \in V \tag{Closure under addition}$$

$$u + v = v + u \tag{Commutative property}$$

$$u + (v+w)=(u+v)+w \tag{Associative property}$$

V has a zero vector 0 such that for every $$\u \in \V$$, $$u+0=u$$. $$\tag{Additive identity}$$

For every $$u \in V$$, there is a vector in V denoted by −u such that u+(−u)=0. $$\tag{Additive inverse}$$

Now let's also assume that $$c,d \in \mathbb R$$

$$cu \in V \tag{Closure under scalar multiplication}$$

$$c(u+v)=cu+cv \tag{Distributive property}$$

$$(c+d)u=cu+du \tag{Distributive property}$$

$$c(du)=(cd)u \tag{Associative property}$$

$$1(u)=u \tag{Scalar identity}$$

 

[fresh_42]

You listed the axioms of a vector space, but how do you define the tangent space? What does an element of $T_pM$ look like?

 

[Delong66]

Definition 1: Suppose M a differentiable manifold and $$p\in M$$.
A funtion $$f:M \rightarrow \mathbb{R}$$ is differentiable at $$p \in M$$ iff $$\exists U_p \subset M$$ : $$f:U_p \rightarrow \mathbb{R}$$ is differentiable.

Definition 2:Dp ={set of all differentiable functions at p in M}

Definition 3: Suppose M a differentiable manifold and $$p\in M$$.
A tangent vector at $$p\in M$$ is an operator $$v:D_p \rightarrow \mathbb{R} \quad : \quad f \mapsto v(f)$$ with
i) $$v(mf+ng)=mv(f)+nv(g) \quad \forall f,g \in D_p$$
ii)$$v(fg)=v(f)g(p)+f(p)v(g) \quad \forall f,g \in D_p$$
\
Definition 4: TpM={set of all tangent vectors at p in M}

 

[fresh_42]

Definitions 1 and 2 define the differentiability at points of $M,$ i.e. differential functions, not their tangents.

Definition 3 defines one tangent vector, definition 4 defines the set of all tangent vectors. However, it does not tell anything about the relation between two such operators $v$ and $w.$ We now have a set, nothing more. How do we make this set a vector space?

We need to define $v+w$ so we set: $(v+w)(f):=v(f)+w(f)$ and $(\alpha v)(f):=\alpha \cdot v(f).$
This makes the set of derivations, the operators, a linear space. The axioms follow automatically.

If you want to calculate something, then consider curves on $M$ through $p$ and calculate actual tangent vectors and show that $D_p(v+w)(f)=D_p(v)(f)+D_p(w)(f).$ You can also use the coordinate directions and partial derivatives:
$$
D_p(v)(f)=\sum_{i=1}^n\, v_i\, \frac{\partial f}{\partial x_i}(p)
$$
This could also help ($p=x_0, J=D, J$ for Jacobi matrix)
https://www.physicsforums.com/insights/pantheon-derivatives-part-ii/