These notes evolved in the process of teaching a beginning graduate course in Hilbert Spaces. The first edition was simply personal, handwritten notes prepared for lecture. A copy was typed and given to the students, for it seemed appropriate that they should have a statement of theorems, examples, and assignments. With each teaching of the course, the notes grew.A text for the course has always been announced, but purchase of the text has been optional. A text provides additional reading, alternate perspectives, and a source of exercises. Whatever text was used, it was chosen with the intent of the course in mind.