The definition of tensor $(E_1, \ldots, E_n)$ are basis for $V$. But this statement makes no sense to me. How could it define $F$ with $F^{j_1 \ldots j_l}_{i_1 \ldots i_k} E_{j_1} \ldots$, but then define $F^{j_1 \ldots j_l}_{i_1 \ldots i_k}$ with $F$? Totally lost...

 A: *

*One presumably has that $\{\varphi^i\}$ is the dual basis of $V^\ast$ defined by $\{E_j\}$, so that
$$
 \varphi^i(E_j) = \delta^i_j.
$$
Hence, $\{E_{j_1} \otimes \cdots \otimes E_{j_l} \otimes \varphi^{i_1} \otimes \cdots \varphi^{i_k}\}$ is a basis of $T^k_l(V) = V^{\otimes l} \otimes (V^\ast)^{\otimes k}$, so that any $F \in T^k_l(V)$ can be written as
$$
 F = F^{j_1\cdots j_l}_{i_1 \cdots i_k} E_{j_1} \otimes \cdots \otimes E_{j_l} \otimes \varphi^{i_1} \otimes \cdots \varphi^{i_k}
$$
for unique constants $F^{j_1\cdots j_l}_{i_1 \cdots i_k}$.

*In light of the canonical isomorphism $V^{\ast\ast} \cong V$, you can identify $T^k_l(V) = V^{\otimes l} \otimes (V^\ast)^{\otimes k}$ as the vector space of all multilinear maps $(V^\ast)^l \times V^k \to F$ via
$$
 (w_1 \otimes \cdots w_k \otimes \psi_1 \otimes \cdots \otimes \psi_l)(f_1,\dotsc,f_k,v_1,\dotsc,v_l) := f_1(w_1) \cdots f_k(w_k) \psi_1(v_1) \cdots \psi_l(v_l)
$$
for $w_m \in V$, $\psi_n \in V^\ast$. Given this, then, you have that
$$
 (E_{j_1} \otimes \cdots \otimes E_{j_l} \otimes \varphi^{i_1} \otimes \cdots \varphi^{i_k})(\varphi^{j_1^\prime},\dotsc,\varphi^{j_l^\prime}, E_{i_1^\prime}, \dotsc, E_{i_k^\prime}) = \delta^{j_1^\prime}_{j_1} \cdots \delta^{j^\prime_l}_{j_l}\delta^{i_1}_{i^\prime_1} \cdots \delta^{i_k}_{i^\prime_k}.
$$

*Thus, for our $F \in T^k_l(V)$,
$$
 F(\varphi^{j_1^\prime},\dotsc,\varphi^{j_l^\prime}, E_{i_1^\prime}, \dotsc, E_{i_k^\prime}) = F^{j_1\cdots j_l}_{i_1 \cdots i_k} E_{j_1} \otimes \cdots \otimes E_{j_l} \otimes \varphi^{i_1} \otimes \cdots \varphi^{i_k}(\varphi^{j_1^\prime},\dotsc,\varphi^{j_l^\prime}, E_{i_1^\prime}, \dotsc, E_{i_k^\prime})  = F^{j_1\cdots j_l}_{i_1 \cdots i_k}\delta^{j_1^\prime}_{j_1} \cdots \delta^{j^\prime_l}_{j_l}\delta^{i_1}_{i^\prime_1} \cdots \delta^{i_k}_{i^\prime_k} = F^{j_1^\prime\cdots j_l^\prime}_{i_1^\prime \cdots i_k^\prime}.
$$
This links to @anon's answer as follows, if you replace "inner product" with "pairing between a vector space and its dual":


*

*Just as for $T^k_l(V)$, $\{\varphi^{j_1} \otimes \cdots \otimes \varphi^{j_l} \otimes E_{i_1} \otimes \cdots \otimes E_{i_k}\}$ can therefore be identified as a basis of $(V^\ast)^{\otimes l} \otimes V^{\otimes k} \cong V^{\otimes k} \otimes (V^\ast)^{\otimes l} =: T^l_k(V)$.

*The canonical isomorphism $V^{\ast\ast} \cong V$ induces a canonical isomorphism 
$$
T^k_l(V) := V^{\otimes l} \otimes (V^\ast)^{\otimes k} \cong (V^{\ast\ast})^{\otimes l} \otimes (V^\ast)^{\otimes k} \cong ((V^\ast)^{\otimes l} \otimes V^{\otimes k})^\ast \cong T^l_k(V)^\ast.
$$
Under this isomorphism, the basis $\{E_{j_1} \otimes \cdots \otimes E_{j_l} \otimes \varphi^{i_1} \otimes \cdots \varphi^{i_k}\}$ of $T^k_l(V)$ can be identified as the dual basis of $T^k_l(V) \cong T^l_k(V)^\ast$ dual to the basis $\{\varphi^{j_1} \otimes \cdots \otimes \varphi^{j_l} \otimes E_{i_1} \otimes \cdots \otimes E_{i_k}\}$ of $(V^\ast)^{\otimes l} \otimes V^{\otimes k} \cong T^l_k(V)$. Thus,
$$
 (E_{j_1} \otimes \cdots \otimes E_{j_l} \otimes \varphi^{i_1} \otimes \cdots \varphi^{i_k})(\varphi^{j_1^\prime} \otimes \cdots \otimes \varphi^{j_l^\prime} \otimes E_{i_1^\prime} \otimes \cdots \otimes E_{i_k^\prime}) = \delta^{j_1^\prime}_{j_1} \cdots \delta^{j^\prime_l}_{j_l}\delta^{i_1}_{i^\prime_1} \cdots \delta^{i_k}_{i^\prime_k}.
$$

*In general, if $W$ is a finite dimensional vector space with basis $\{e_j\}$, and if $\{\varepsilon^j\}$ is the dual basis of $W^\ast$ defined by $\{e_j\}$, i.e., $\varepsilon^j(e_k) = \delta^j_k$, then for any $f \in W^\ast$, writing $f = f_j \varepsilon^j$, we have that $f(e_k) = f_j \varepsilon^j(e_k) = f_j \delta^j_k = f_k$.

*Putting everything together, we therefore have that $F \in T^k_l(V)$ can be written as
$$
 F = F^{j_1\cdots j_l}_{i_1 \cdots i_k} E_{j_1} \otimes \cdots \otimes E_{j_l} \otimes \varphi^{i_1} \otimes \cdots \varphi^{i_k}
$$
for unique constants $F^{j_1\cdots j_l}_{i_1 \cdots i_k}$, so that
$$
 F(\varphi^{j_1^\prime} \otimes \cdots \otimes \varphi^{j_l^\prime} \otimes E_{i_1^\prime} \otimes \cdots \otimes E_{i_k^\prime})\\ = F^{j_1\cdots j_l}_{i_1 \cdots i_k} E_{j_1} \otimes \cdots \otimes E_{j_l} \otimes \varphi^{i_1} \otimes \cdots \varphi^{i_k}(\varphi^{j_1^\prime} \otimes \cdots \otimes \varphi^{j_l^\prime} \otimes E_{i_1^\prime} \otimes \cdots \otimes E_{i_k^\prime})\\  = F^{j_1\cdots j_l}_{i_1 \cdots i_k}\delta^{j_1^\prime}_{j_1} \cdots \delta^{j^\prime_l}_{j_l}\delta^{i_1}_{i^\prime_1} \cdots \delta^{i_k}_{i^\prime_k} = F^{j_1^\prime\cdots j_l^\prime}_{i_1^\prime \cdots i_k^\prime}.
$$
