I have two questions about Spivak's Calculus on Manifolds.

$(1)$ In page 77 he proves that the set of k-tensor products of elements of a dual basis is linearly independent in the following way:

Suppose now that there are numbers $a_{i_1}\dots _{i_k}$ such that $\sum_{i_1\dots ik}^n a_{i_1}\dots _{i_k}\cdot\varphi_{i_1}\otimes\dots \otimes\varphi_{i_k}=0$. Applying both sides of this equation to $(v_{j_1},\dots, v_{j_k})$ yields $a_{i_1}\dots _{i_k}=0$.

$\varphi$ are basic elements of the dual basis and $\varphi_i(v_j)=\delta_{ij}$.

I don't understand how the proof works. To prove the linear independence of $\varphi_{i_1}\otimes\dots \otimes\varphi_{i_k}$ he needs to show that the scalars of the linear combinations are zero, but why does he apply "both" sides to $(v_{j_1},\dots, v_{j_k})$? And what does "apply to both sides" mean? He apparently applies the vector $(v_{j_1},\dots, v_{j_k})$ to the right side. I suppose that $0$ on the right side is not the number zero but a function zero?

$(2)$ In the page 77 he says

If $f:V\rightarrow W$ is a linear transformation, a linear transformation $f^*:\tau^k(W)\rightarrow \tau^k(V)$ is defined by $f^*T(v_1,\dots,v_k)=T(f(v_1),\dots, f(v_k))$ for $T\in\tau^k(W)$, $v_1,\dots,v_k\in V.$

$\tau^k(V)$ and $\tau^k(W)$ are the sets of all k-tensors on $V$ and $W$.

My problem here: What does the left side mean? If is to be understood as $f^*(T(v_1,\dots,v_k))$ it wouldn't make any sense because $T$ takes elements of $W$ to $\mathbb{R}$ ($T\in\tau^k(W)$) - the right side works because $T$ is taking the $f(v_i)$'s. So what is the $\tau^k(W)$ element $f^*$ takes to $\tau^k(V)$?


For your first question, Spivak means that you input the elements $(v_{j_1},...,v_{j_k})$ into the tensor product (which is a linear functional on the space $V\otimes V\otimes...\otimes V$ where $v_{j_t}\in V$) and also into the other side, $0$, which is the functional taking any element $(v_{j_1},...,v_{j_k})$ to zero. By choosing different values of $j_1,...,j_k$ and using $\phi_i(v_j)=\delta_{ij}$, we can cancel out all but one term in the sum on the left-hand side, and therefore get $a_{j_1...j_k}=0$ (for whichever $j_1,...,j_k$ we have chosen).

For your second question, Spivak is telling you what $f^*T$ does to $(v_1,...,v_k)\in V\otimes...\otimes V$. Namely, it takes the functional $T\in\tau^k(W)$, which is defined by whatever it does to elements of $W\otimes...\otimes W$, and maps it to the functional $f^*T$, which does the following to elements of $V\otimes...\otimes V$: first it maps them to $W\otimes...\otimes W$ by mapping each component of a primitive tensor to $f$ applied to that component; then it takes $T$ of the result.


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