  In our previous work [Singler, SIAM J. Numer. Anal. 52 (2014), no. 2,
852-876], we considered the proper orthogonal decomposition (POD) of time
varying PDE solution data taking values in two different Hilbert spaces. We
considered various POD projections of the data and obtained new results
concerning POD projection errors and error bounds for POD reduced order models
of PDEs. In this work, we improve on our earlier results concerning POD
projections by extending to a more general framework that allows for
non-orthogonal POD projections and seminorms. We obtain new exact error
formulas and convergence results for POD data approximation errors, and also
prove new pointwise convergence results and error bounds for POD projections.
We consider both the discrete and continuous cases of POD. We also apply our
results to several example problems, and show how the new results improve on
previous work.
