- #1
yifli
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Homework Statement
Let {M_i} be an orthogonal sequence of complete subspaces of a pre-Hilbert space V, and let P_i be the orthogonal projection on M_i. Prove that {P_i(e)} is Cauchy for any e in V
2. The attempt at a solution
I'm trying to prove as n and m goes infinity, [tex]\left\|P_n(e)-P_m(e)\right\|^2\rightarrow 0[/tex]
Here is what I've got so far:
[tex]\left\|P_n(e)-P_m(e)\right\|^2=\left\|P_n(e)\right\|^2+\left\|P_m(e)\right\|^2[/tex] because P_n(e) is orthogonal to P_m(e), how to proceed?