What is the norm for the product of normed spaces?

Question

Suppose $(\Omega, \Sigma ,\lambda)$ is a probability space and $X_i=L^2(\Omega,\Sigma,\lambda ,[0,1])$ with norm $||.||_{L^2}$ for all $i\in I$, $I$ is finite.

1. Is there any natural norm for the product space $X=\prod_{i\in I} X_i$?

2. Is it possible to generate any norm on $X$ from an inner product $(. |.)： X \times X\to R$, i.e.

$$(x|y)= \sum_{i\in I} \int_{\Omega } x_i y_i d\lambda$$

and

$$ ||x|| = \sqrt{\sum_i ||x||^2_{L^2}}=\sqrt{\sum_{i\in I} \int_{\Omega } x_i^2 d\lambda }$$?

3. If now I define a vector-valued $L^2$ space $Y=L^2(\Omega,\Sigma,\lambda,[0,1]^I)$ with the norm

$$||y||_Y= \sqrt{\int_{\Omega } \sum_{i\in I} y_i^2 d\lambda }$$

Is the spaces $Y$ and $X$ equivalent in the sense that each element in $Y$ uniquely corresponds to an element in $X$?

Thanks.