## Elementary Linear Algebra 7th Edition

Let $u=(1,1,1)$, then we have $$\langle u, u\rangle=-u_1u_2u_3\Longrightarrow \langle (1,1,1), (1,1,1)\rangle=-1<0.$$ This shows that the function does not define an inner product.
Let $u=(1,1,1)$, then we have $$\langle u, u\rangle=-u_1u_2u_3\Longrightarrow \langle (1,1,1), (1,1,1)\rangle=-1<0.$$ This shows that the function does not define an inner product.