Given the mathematical formalization of the statistical regression problem, let $ThetasubseteqGamma$ be a set of coefficients. The hypothesis of the linear regression is:

$exists\; (beta^0,cdots,beta^p)intheta^\{p+1\}:$ $mathbb\{E\}(Y|X\_1,cdots,X\_p)=beta^0\; +\; sum\_\{j=1\}^p\; beta^j\; X\_j$

and the metric used is:

$forall\; f,gin\; F,\; d(f,g)\; =\; mathbb\{E\}[(f-g)^2]$

We therefore want to minimize $mathbb\{E\}[(Y-f(X\_1,cdots,X\_p))^2]$, which means that

$f(X\_1,cdots,X\_p)=mathbb\{E\}(Y|X\_1,cdots,X\_p)\; =\; beta^0\; +\; sum\_\{j=1\}^p\; beta^j\; X\_j$

Hence, we only need to find $beta^0,cdots,beta^p$.