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A square integrable function on $\mathbb{R}^n$ would be defined as a function $f(\mathbf{x})$ satisfying the constraint $$ \int |f(\mathbf{x})|^2 d^n\mathbf{x} < \infty, $$ for $\mathbf{x} \in \mathbb{R}^n$. Is there a specific name given to the quantity $$ \int |f(\mathbf{x})|^2 d^n\mathbf{x}? $$

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    $\begingroup$ Isn't that the $\ell_2$ norm for $f$ or or something? $\endgroup$
    – Asaf Karagila
    Apr 9, 2015 at 13:20

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This is the square of a quantity normally called the $2$-norm, or the $L^2$-norm. As its name suggests, it is a norm on the space of square-integrable functions, which is called $L^2(\mathbb{R}^n)$.

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    $\begingroup$ ... the square of the $L^2$-norm, the norm itself is given by $$ \|f\|_{L^2} = \left(\int_{\mathbf R^n} |f(x)|^2\, dx\right)^{1/2} $$ $\endgroup$
    – martini
    Apr 9, 2015 at 13:25
  • $\begingroup$ @martini Fair point... $\endgroup$
    – Chappers
    Apr 9, 2015 at 13:26
  • $\begingroup$ @martini Good spot! $\endgroup$
    – Rammus
    Apr 9, 2015 at 13:29

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