Best-approximation with tensors of rank $ge2$

Let $kinmathbb N$, $H_i$ be a (finite-dimensional, if necessary) $mathbb R$-Hilbert space for $iin I:={1,ldots,k}$, $H:=bigotimes_{iin I}H_i$ denote the tensor product$^1$ of $(H_i)_{iin I}$ and $$H^{(r)}:=left{uin H:operatorname{rank}u=rright};;;text{for }rinmathbb N_0.$$

I’m struggling to understand the importance and implication of the following result: Let $vin H$.

1. There is a $uin H$ with $operatorname{rank}u=1$ and $$left|u-vright|_H=inf_{uin H^{(1)}}left|u-vright|_Htag1.$$
2. There is a $uin H^{(3)}$ and a $(u_n)_{ninmathbb N}subseteq H^{(2)}$ with $$left|u_n-uright|_Hxrightarrow{ntoinfty}0tag2.$$

Okay, by 1., there is a (not necessarily unique) minimizer of $$H^{(1)}to[0,infty);,;;;umapstoleft|u-vright|_Htag3.$$

Question 1: But why can we infer from 2. that the analogous problem of minimizing $$H^{(2)}to[0,infty);,;;;umapstoleft|u-vright|_Htag4$$ may have no solution? I guess we need to take $v=u$ (with $u$ as in 2.), but why does the existence of $(u_n)_{ninmathbb N}$ imply that there is no solution?

Question 2: That we can only guarantee the existence of a minimizer of $$H^{(r)}to[0,infty);,;;;umapstoleft|u-vright|_Htag5$$ for $r=1$ is unsatisfactory only if it would actually be beneficial to take $r$ as large as possible. I could imagine that the error $left|u^{(r)}-vright|_H$ of a hypothetical minimizer $u^{(r)}$ of $(5)$ is nonincreasing in $rinmathbb N_0$. Is this the case? If so, how can we show this?

Question 3: Can we infer from 2. that $(5)$ may have no minimizer for all $rge2$?

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$^1$ If $E_i$ is a $mathbb R$-vector space, I’m defining $$(x_1otimes x_2)(B):=B(x_1,x_2);;;text{for }Binmathcal B(E_1times E_2)text{ and }x_iin E_i,$$ where $mathcal B(E_1times E_2)$ is the space of bilinear forms on $E_1times E_2$, and $$E_1otimes E_2:=operatorname{span}{x_1otimes x_2:E_iin E_i}subseteq{mathcal B(E_1times E_2)}^ast.$$