I'm studying about tensors and have already understood the following theorem:
$C^1_1$ is the unique linear function such that $C_1^1(v\otimes\eta)=\eta(v)=v(\eta)$ for all $v\in V$ and $\eta\in V^*$.
Now I'm reading about this theorem:
Let $1\leq k\leq r, 1\leq l\leq s$. Then there exists a unique linear map $C^k_l:\mathcal{T}^r_s(V)\to\mathcal{T}^{r-1}_{s-1}(V)$ such that
$$C^k_l(v_1\otimes\ldots\otimes v_r\otimes\eta^1\otimes\ldots\otimes\eta^s)=v_k(\eta^l)v_1\otimes\ldots\otimes \hat v_k\otimes\ldots\otimes v_r\otimes\eta^1\otimes\ldots\otimes\hat\eta^l\otimes\ldots\otimes\eta^s$$
where the hat indicates that that term is omitted.
The proof given in the book is kind of unintuitive to me, so I tried an alternative method of proving it. Here's my attempt:
Let's define $$C^k_l(A)\equiv A^{i_1\ldots i_r}_{j_1\ldots j_s}\ C_1^1(h_{i_k}\otimes\theta^{j_l})\ h_{i_1}\otimes\ldots\otimes \hat h_{i_k}\otimes\ldots\otimes h_{i_r}\otimes\theta^{j_1}\otimes\ldots\otimes\hat\theta^{j_l}\otimes\ldots\otimes\theta^{j_s}$$ where $h_i$ and $\theta^j$ are basis vectors and covectors respectively.
With the above definition,
$$C^k_l(v_1\otimes\ldots\otimes v_r\otimes\eta^1\otimes\ldots\otimes\eta^s)$$
$$=(v_1)^{i_1}\ldots (v_r)^{i_r}(\eta^1)_{j_1}\ldots (\eta^s)_{j_s}C_1^1(h_{i_k}\otimes\theta^{j_l})\ h_{i_1}\otimes\ldots\otimes \hat h_{i_k}\otimes\ldots\otimes h_{i_r}\otimes\theta^{j_1}\otimes\ldots\otimes\hat\theta^{j_l}\otimes\ldots\otimes\theta^{j_s}$$
Pulling $(v_k)^{i_k}$ and $(\eta^l)_{j_l}$ into the $C_1^1$ argument and the rest of the components into the big tensor product on the right, we get the RHS of the desired result.
Uniqueness follows from the uniqueness of $C_1^1$.
Are there any issues with the above proof? Would appreciate any help!
Best Answer
Any linear map $C$ between two vector spaces ${\cal U}$ and ${\cal W}$ is uniquely defined as long as it is defined on a basis of ${\cal U}\,.$ A basis of ${\cal T}^r_s(V)$ is $$\tag{1} h_{i_1}\otimes\,...\,\otimes\,h_{i_r}\otimes \theta^{j_1}\,\otimes\,...\otimes\,\theta^{j_s} $$ where all $i_k$ and all $j_l$ run through $\{1,...,n\}$ and $\{h_1,...,h_n\}$ is a basis of $V$ and $\{\theta_1,...\theta_n\}$ is a basis of $V^*\,.$ Unfortunately I do not have access to Newman's book but I strongly believe that he defines the map $C^k_l$ on a larger set than a basis of ${\cal T}^r_s(V)$ by allowing each of the factors $v_i$ and $\eta^j$ to be a linear combination of the basis vectors $h_\mu\,,$ resp. $\theta_\nu\,.$ (If his $v_i$ were all different, or his $\eta^j\,,$ this would not define $C^k_l$ on a basis of ${\cal T}^r_s(V)\,.$ Clearly, in (1) the factors are not all different.)
The only thing (if anything) that needs to be checked is if the contraction form of the right hand side of \begin{align} &C^k_l(v_1\otimes\ldots\otimes v_r\otimes\eta^1\otimes\ldots\otimes\eta^s)\\&=v_k(\eta^l)\,v_1\otimes\ldots\otimes \hat v_k\otimes\ldots\otimes v_r\otimes\eta^1\otimes\ldots\otimes\hat\eta^l\otimes\ldots\otimes\eta^s\tag{2} \end{align} is compatible with the same form for basis vectors: \begin{align} &C^k_l(h_{i_1}\otimes\,...\,\otimes\,h_{i_r}\otimes \theta^{j_1}\,\otimes\,...\otimes\,\theta^{j_s})\\&=h_{i_k}(\theta^{j_l})\,h_{i_1}\otimes\ldots\otimes \hat h_{i_k}\otimes\ldots\otimes h_{i_r}\otimes\theta^{j_1}\otimes\ldots\otimes\hat\theta^{j_l}\otimes\ldots\otimes\theta^{j_s}\,.\tag{3} \end{align} By linearity it is enough to expand only the hatted factors $$ v_k=\alpha{^\mu}_k\,h_\mu\,,\quad\eta^l={\beta^l}_\nu\,\theta^\nu\quad\text{ (using summation convention) } $$ and assume that in (2) the un-hatted factors are basis vectors. In other words we need to check that (3) implies \begin{align} &C^k_l(h_{i_1}\otimes\ldots\otimes v_k\otimes\ldots\otimes h_{i_r}\otimes\theta^{j_1}\otimes\ldots\otimes \eta^l\otimes\ldots\otimes\theta^{j_s})\\&=v_k(\eta^l)\,h_{i_1}\otimes\ldots\otimes \hat{v}_k\otimes\ldots\otimes h_{i_r}\otimes\theta^{j_1}\otimes\ldots\otimes \hat{\eta}^l\otimes\ldots\otimes\theta^{j_s}\,.\tag{2'} \end{align} From $$ v_k(\eta^l)=\alpha{^\mu}_k{\beta^l}_\nu\,h_\mu(\theta^\nu) $$ this should however by obvious. If not please note that $\mu,\nu$ are just dummy indices which we can replace: $$ v_k(\eta^l)=\alpha{^{i_k}}_k{\beta^l}_{j_l}\,h_{i_k}(\theta^{j_l})\,. $$