I'm trying to understand he basic notation(s) used to write out tensors, namely \begin{equation} T = T^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n} \frac{\partial}{\partial x^{\nu_1}} \otimes...\otimes \frac{\partial}{\partial x^{\nu_m}} \otimes \mathrm{d}x^{\mu_1} \otimes ... \otimes \mathrm{d}x^{\mu_n} \quad (1) \end{equation} which defines a $(m,n)$-tensor as a multilinear map $T:\otimes_{m} T^{*}_pM \otimes_{n} T_pM \to \mathbb{R}$, and \begin{equation} T(\omega_1,...,\omega_m,V_1,...,V_n) = T^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n} \omega_{1,\mu_1}...\omega_{m,\mu_m} V_{1}^{\nu_1}...V_{n}^{\nu_n} (2) \end{equation} which (according to book by Nakahara (it's for physicists)) defines an action of T on $\omega$ and $V$, where $\omega_i = \omega_{i,\mu} \mathrm{d}x^{\mu}$ (cotangent vector i.e. one-form), $V_{j}=V_{j,\nu} \frac{\partial}{\partial x^{\nu}}$ (tangent vector), $1 \leq i \leq m$ and $1 \leq j \leq n$.
First question. In (1) are $T^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n}$ the components of the tensor in the basis in question? If yes, the notation is slightly odd, considering $T$ is a function. Seemingly a more sensible notation would be something like $T(v_{\nu_1},...,v_{\nu_m},v_{\mu_1},...,v_{\mu_n}) = r^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n}$, where $v$ is an element of the direct product space, and $v_{i}$ are it's components. On the other hand if $T^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n}$ is not the component, then what is it?
Second question. What is the second definition (2) trying to say? As tensor was just now defined as a function (instead of the elements of the product space themselves), I understand that $T$ gives out a real number from an element of the product space, but then what is $T^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n}$ here, or where does it come from? Looking at (2) I think I've misunderstood $T^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n}$.
I was thinking that example(s) would help. Let $M=\mathbb{R}^3$, an $\omega_1 = \omega_{1,x} \mathrm{d}x + \omega_{1,y}\mathrm{d}y$, $\omega_2 = \omega_{2,y} \mathrm{d}y + \omega_{2,z}\mathrm{d}z$ and $V = V_{x} \partial_x + V_{y} \partial_y + V_{z} \partial_z$. If I want to take a tensor product $\omega_1 \otimes \omega_2 \otimes V$, I should get a tensor with $3^3=27$ components (or that's what wolfram says). So according to (1) I'd think that the tensor I'd get would be $T = \omega_{1,\mu_1} \omega_{2,\mu_"} V_{\nu} \partial_{\nu} \otimes \mathrm{d}x^{1,\mu_1} \otimes \mathrm{d}x^{2,\mu_2}$ with components $\omega_{1,\mu_1} \omega_{2,\mu_2} V_{\nu}$? (I understand that every tensor can't be constructed this way).
In 2) you have how much is the value of $T$ at the arguments $v_{\mu_1},...,V_n$.