Is $\sum_i x_i \sum_j y_j$ a valid inner product for $\mathbb{R}^n$?

Question

An exercise in my book asks you to check whether the following is a valid inner product for $\mathbb{R}^n$: $$ \left<x,y\right>=\sum_{i=1}^{n}x_i \sum_{j=1}^{n}y_j $$ The answer given is "No", but to me it seems to satisfy all the axioms (see below).

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\begin{array} .\left<y,x\right> &= \sum_i y_i \sum_j x_j\\ &= \sum_i x_i \sum_j y_j\\ &= \left<x,y\right> \tag{1} \end{array}

\begin{array} .\left<x,y+z\right> &= \sum_i x_i \sum_j(y_j+z_j)\\ &= \sum_i x_i \left(\sum_j y_j + \sum_j z_j\right)\\ &= \sum_i x_i \sum_j y_j + \sum_i x_i \sum_j z_j\\ &= \left<x,y\right> + \left<x,z\right> \tag{2} \end{array}

\begin{array} a\left<cx,y\right> &= \sum_i (cx_i)\sum_j y_j\\ &= c\sum_i x_i \sum_j y_j\\ &= c\left<x,y\right> \tag{3} \end{array}

\begin{array} a\left<x,x\right>&=\sum_i x_i \sum_j x_j\\ &=\left(\sum_i x_i\right)^2\\ &>0 \quad \textrm{if }x\neq 0 \tag{4} \end{array}

Answer 1

For $(4)$ you say $<x,x> = (\sum _{i}x_i)^2>0$ if $x\neq 0.$

However try $x=(1,-1,0,\cdots,0)$.