I'm reading a paper from M. Hairer on Malliavin calculus. I have a question regarding a formula in the middle of page 6. hereunder an extract :
Let $H$ be a Hilbert space with orthonormal basis $(e_i)_{i \in \mathbb{N}}$. A multiindex k is viewed as a function $k : \mathbb{N} \to \mathbb{N}$ such that all but finitely many values vanish.
Write now $H^{\otimes_s n}$ for the subspace of $H^{\otimes n}$ consisting of symmetric tensors. There is a natural projection $\Pi: H^{\otimes n} \to H^{\otimes_s n}$ given as follows. For any permutation $\sigma$ of $\{1, . . . , n\}$ write $\Pi_\sigma: H^{\otimes n} \to H^{\otimes n}$ for the linear map given by $$ \Pi_\sigma(h_1\otimes ...\otimes h_n) = h_{\sigma(1)}\otimes ...\otimes h_{\sigma(n)} $$
We then set $\Pi = \frac{1}{n!}\sum_\sigma \Pi_\sigma$, where the sum runs over all permutations. Writing $|k|=\sum_i k_i$, we set $e_k = \Pi \bigotimes_i e_i ^{\otimes k_i}$, which is an element of $H^{\otimes_s |k|}$ . Note that the vectors $e_k$ are not orthonormal, but that instead one has $$ \langle e_k, e_l \rangle = \frac{k!}{|k|!}\delta_{k,l} $$
I'm having difficulties understanding this last formula for the case where $|k| \neq |l|$ take for instance $k=(1,0,...)$ and $l=(0,2,0,...)$. Then $e_k=e_1$ and $e_l=e_2 \otimes e_2$. How would one define $\langle e_1, e_2 \otimes e_2 \rangle$ ? Should we just claim that $\langle e_k, e_l \rangle = 0$ whenever $|k| \neq |l|$ ?
Thanks for you help