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I am referring to a proof in Part II of Rudin's Functional Analysis.

I got confused by his proof of Thm 6.26 (page 167). He says by applying (2) successively we can get inequality (4), but I do not see how this can be done. If I am right, $T^N$ should be the operator $(D_1D_2D_3\dots D_n)^N$. I do not see how this is related to the $N$-norm of a function.

Can someone elaborate a little bit?

Thanks!

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The estimate (2) says that you can keep adding derivatives to the right hand side. For example, if you want to estimate $D^\alpha\psi$ with $|\alpha|=N$, then you can do $ |D^\alpha\psi| \leq \max_Q |D_1^{\alpha_1}\ldots D_k^{\alpha_k+1}\ldots D_n^{\alpha_n}\psi|. $ Now if you want to have the orders of all partial derivatives in the right hand side equal to each other, in the worst case, such as $\alpha=(N,0,\ldots,0)$, you need to have $N$ derivatives in each direction.