Reddit reviews The Cauchy-Schwarz Master Class (MAA Problem Books)

I've heard nothing but good things about this book.

You might try the aptly named book Introduction to Hilbert Spaces. I haven't spent much time with it myself (I slogged through Conway at the beginning of grad school), but it looks good for your application.

Also, I was reminded of this cool book:
Cauchy-Schwarz masterclass.

The Cauchy-Schwarz Master Class is a great book on inequalities that will really improve your understanding of how and when to apply specific techniques. Highly recommended, and the paperback version isn't too overpriced.

That being said, competition mathematics also requires that you be able to recognize which technique to use (and then do it) quickly. Improving your speed is really accomplished by doing problem sets, and I don't have a good collection to suggest here.

Thanks!

I'm working through _The Cauchy-Schwartz Master Class_, and never cease to be amazed how far the CS inequality gets. But I never thought of trying that. (Truth be told, I didn't try past my long commute)

I would also recommend Cauchy-Schwarz Master Class. It has a lot of interesting problems without requiring a whole lot of prerequisite equipment. It also does an incredible job explaining how to go about solving problems.
http://www.amazon.com/The-Cauchy-Schwarz-Master-Class-Introduction/dp/052154677X

Not sure what your background is, but I quite liked Steele's book.