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'''Step 2. the General Case'''
Let <math>\rho _\epsilon</math> be a family of [[Mollifier|mollifiers]], <math>f_\epsilon=f \ast \rho_\epsilon</math>, then for each ε, <math>f_\epsilon'=f\ast \rho_\epsilon'</math>, <math>|f_\epsilon'|=|f\ast \rho_\epsilon'|\leq \|f\|_{L^2}\|\rho_\epsilon'\|_{L^2}</math> by [[Hölder's inequality]], hence <math>f_\epsilon</math> is differentiable and has a bounded derivative. By '''Step 1''', <math>\int |f_\epsilon(x)|^2 dx=\int |\hat{f_\epsilon }(\xi)|^2 d\xi</math>. By the property of mollification, the left hand side converges to <math>\|f\|_{L^2}</math> as <math>\epsilon\rightarrow 0</math>, and by [[convolution theorem]], <math>|\hat{f_\epsilon }|=|\hat{f}||\hat{\rho_\epsilon }|\rightarrow |\hat{f}| \text{ as }\epsilon\rightarrow 0 </math>, hence <math>\liminf\limits_{\epsilon\rightarrow 0}|\hat{f_\epsilon}|=|\hat{f}|</math>, by MCT, we have <math>\int |\hat{f}|^2 d\xi=\int \liminf \limits_{\epsilon\rightarrow 0} |\hat{f_\epsilon}|^2 d\xi = \liminf \limits_{\epsilon\rightarrow 0}\int |\hat{f_\epsilon}|^2 d\xi = \liminf \limits_{\epsilon\rightarrow 0} \int |f_\epsilon|^2 dx =\int |f|^2 dx </math>
== Extensions ==
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