I need some assistance in finding ||x^2|| using both the Laguerre and the Legendre inner products.
So in your question, I guess if x=(x,y) then $\displaystyle x^2 = (x^2,y^2) $. You know that if $\displaystyle x^2$ is an inner product space the norm is defined by $\displaystyle \|x^2\| = \sqrt{\left\langle x^2,y^2 \right\rangle}$.