What is the norm on $l^2 \times l^2$

Question

Suppose I have the normal norm for $l^2$. How is the norm for $l^2 \times l^2$ usually defined? Since I am on a Hilbert space over $l^2 \times l^2$, is the norm naturally induced by some inner product?

Thanks.

Answer 1

If $E$ and $F$ are inner product spaces, then it is standard to define an inner product on $E\times F$ (which is usually written $E\oplus F$) by $$\langle(e,f),(e',f')\rangle=\langle e,e'\rangle+\langle f,f'\rangle.$$ The idea is that this inner product treats $E$ and $F$ as being orthogonal to each other inside $E\oplus F$ (identifying $e\in E$ with $(e,0)$ and $f\in F$ with $(0,f)$). The induced norm is then given by $$\|(e,f)\|^2=\|e\|^2+\|f\|^2.$$