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In the chapter on energy methods for partial differential equations I saw the following: $$\frac{d\|u\|_2^2}{dt}=(u,u_t)+(u_t,u)=\cdots$$ So, why we can't just write $$\frac{d\|u\|_2^2}{dt}=2(u,u_t)=\cdots?$$

Are these two the same, or there is a reason I have to consider the derivative as in the first expression?

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For real valued functions, the two are indeed the same. For complex valued functions, the inner product is sesquilinear and Hermitian (not necessarily symmetric), so complex conjugation comes into play.

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