What shown belove is a reference from Analysis on Manifolds by James Munkres
So we observe that we can structure the set $\Omega^k(V)$ of $k$-tensor field defined over the open set $V$ of $\Bbb R^n$ to a vector space letting $$ [\omega+\eta](x):=\omega(x)+\eta(x)\,\,\,\text{and}\,\,\,[a\cdot\omega](x):=a\cdot\omega(x) $$ for any $x\in V$ where $\omega$ and $\eta$ are $k$-tensor field and $a$ is a scalar. So we define now a function $⊛$ from $\underset{k\,\,\,\text{times}}{\underbrace{\Omega^1(V)\times...\times\Omega^1(V)}}$ to $\Omega^k(V)$ through the equation $$ [\omega_1\circledast...\circledast\omega_k](x):=\omega_1(x)\otimes...\otimes\omega_k(x) $$ for any $x\in V$. So since the collection $$ \mathfrak F:=\big\{\tilde{\phi}_{i_1}(x)\otimes...\otimes\tilde{\phi}_{i_k}(x):i_1,...,i_k=1,...,n\big\} $$ is for each $x\in V$ a base for $\mathcal L^k\big(\mathcal T_x(\Bbb R^n)\big)$ then the tensor field of the colletion $$ \mathfrak T:=\big\{\tilde{\phi}_{i_1}\circledast...\circledast\tilde{\phi}_{i_k}(x):i_1,...,i_k=1,...,n\big\} $$ are linearly independent but unfortunately I did not be able to prove that they span $\Omega^k(V)$ so that I thought that the dimesion of this space is infinite. So I ask to find the dimension of the space of tensor fields defined over an open set $V$ either over a manifold $M$. So could someone help me, please?