I am currently trying to understand tensors and tensor product spaces better, but I find it challenging to recognize some of the subtleties in this area. In particular, I am often unsure about uniqueness statements involving tensors. I would appreciate help in clarifying the uniqueness of certain maps in the following example:
Let $H$ be a base complete inner product space, i.e. a Hilbert space, over $\mathbb{K}$ and define $T^3 = H\otimes H\otimes H$. Let $v\in H$ be fixed and define
$$f_v(e_1,e_2,e_3) := p_v\left(t(e_1,e_2,e_3)\right) = \left<e_3,v\right>e_1\otimes e_2$$
for
$$t:H^3\to T^3, t(e_1,e_2,e_3) := \bigotimes_{k=1}^3e_k$$
where $H^3 = H\times H\times H$, and
$$p_v(\bigotimes_{k=1}^3e_k) := \left<e_3,v\right>e_1\otimes e_2$$
when $a\otimes b$ is a $\mathbb{K}$-multilinear map such that $(a\otimes b)(v_1,v_2) = \left<a,v_1\right>\left<b,v_2\right>, v_1,v_2\in H$.
Take it as granted that $p_v$ is continuous over $T^3$. Is it now true that $p_v$ is unique, while $f_v$ is not? If so, how could we make both of them unique?