- #1

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Orthonormal Basis Expression for Ordinary Contraction: Let ##(V,g)## be a scalar product space with signature ##(\epsilon_1, \ldots, \epsilon_n)##, and let ##(e_1,\ldots,e_n)## be an orthonormal basis for ##V##. If ##A\in\mathcal{T}^1_s(V)## (where ##s\geq 2)## and ##l<s##, then the tensor ##C^1_l(A)\in\mathcal{T}^0_{s-1}(V)## is given by: $$C^1_l(A)(v_1,\ldots,v_{s-1})=\sum_i\epsilon_i\langle\mathcal{R}^{-1}_s(A)(v_1,\ldots,v_{l-1},e_i,v_{l+1},\ldots,v_{s-1}),e_i\rangle$$

For context, ##\mathcal{R}_s:Mult(V^s,V)\to\mathcal{T}^1_s## is the representation map, which acts like this:

$$\mathcal{R}_s(\Psi)(\eta,v_1,\ldots,v_s)=\eta(\Psi(v_1,\ldots,v_s))$$

I don't understand the importance of the relation that the author is trying to prove here - it seems kind of irrelevant. Judging from the name of the theorem, shouldn't it be sufficient to figure out ##C^1_l(A)(e_{j_1},\ldots,e_{j_{s-1}})##, i.e. the components of the contracted tensor w.r.t. different combinations of orthonormal basis vectors? And indeed this is what the author does in the first 2 lines of the proof.

But why are we going overboard to get the result in the theorem statement? Does it mean something significant?