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Let $\{e_n\}$ be an orthonormal basis of a Hilbert space $H$ and $P_n$ be the orthogonal projection onto span$\{e_1,....,e_n\}$. Show that for any bounded linear operator $T$ : $H$ $\longrightarrow$ $H$ and $h$ $\in$ $H$, we have $P_nTP_nh$ $\longrightarrow$$Th$. I was trying to show that $||P_nTP_nh - Th||^2$ $\longrightarrow$ $0$, but to no avail. Thanks for any help.

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    $\begingroup$ Write $$Th - P_n T P_n h = (Th - P_n Th) + P_nT(h - P_nh)$$ and show that each of the two terms converges to $0$. $\endgroup$ – Daniel Fischer Jun 15 '17 at 12:56
  • $\begingroup$ Thanks. That's very helpful. $\endgroup$ – Ester Jun 15 '17 at 13:03
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Let $N_n = 1 - P_n$ so that $h = P_n h + N_n h$. Show that $P_n h \to h$ and $N_n h \to 0$.

Now $$Th = P_n Th + N_n Th = P_n T (P_n h + N_n h) + N_n Th \\ = P_n T P_n h + P_n T N_n h + N_n T h$$ so $$\| Th - P_n T P_n h \| = \| P_n T N_n h + N_n T h \| \leq \| P_n T N_n h \| + \| N_n T h \|.$$

Here $$\| P_n T N_n h \| \leq \| T N_n h \| \leq \|T\| \, \| N_n h \| \to 0 $$ and $$\| N_n T h \| \to 0$$ so $$\| Th - P_n T P_n h \| \to 0$$

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