Dr. Miller has tipped me off that the suggested text for his upcoming course An Introduction to Hilbert Spaces will be:

Berberian, Sterling Khazag. Introduction To Hilbert Space. Oxford University Press, 1961. Reprint Literary Licensing, 2012.

He’s not happy that it ignores measure theory as a means to introduce the Lebesque integral, so he’ll be supplementing that with additional notes. I’ve ordered a used copy of the 1st edition, but there are also versions from AMS as well as a more recent reprint from 2012.

He also suggested that Debnath & Mikusinski was pretty good, albeit more expensive than he would like in addition to not being a fan some of their approaches to topics.

Debnath, Lokenath, and Piotr Mikusinski. Introduction to Hilbert Spaces with Applications. 3rd ed., Academic Press, 2005.