Question
Let $f(x)$ be a function, and $\|f\|$ its $\mathrm{L}^2$ norm on $[a, b]$. Is $\left\|f^2\right\|=\|f\|^2$ ? If yes, prove the statement. If no, give a counterexample.
Step 1
The $\mathrm{L}^2$ norm of $f$, denoted $\|f\|$, is given by \[ \|f\| = \left(\int_a^b |f(x)|^2 \, dx\right)^{1/2}. \] Show more…
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