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Measure Theory and Functional Analysis
Notes 30.4 Review Questions
1. If P and Q are the projections on closed linear subspaces M and N of H, prove that PQ is a
projection PQ QP. In this case, show that PQ is the projection on M N.
2. If P and Q are the projections on closed linear subspaces M and N of H, prove that the
following statements are all equivalent to one another:
(a) P Q;
(b) Px Qx for every x;
(c) M N;
(d) PQ P;
(e) QP = P.
3. If P and Q are the projections on closed linear subspaces M and N of H, prove that Q P is
a projection P Q. In this case, show that Q – P is the projection on N M .
30.5 Further Readings
Books Borbaki, Nicolas (1987), Topological Vector Spaces, Elements of mathematics, Berlin:
Springer – Verlag
Rudin, Walter (1987), Real and Complex Analysis, McGraw-Hill.
Online links www.math.Isu.edu/~ sengupta/7330f02/7330f02proiops.pdf
Planetmath.org/....OrthogonalProjections OntoHilbertSubspaces.html.
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