What makes Hilbert spaces so useful in signal processing? In modern signal processing, we often represent a signal as a point in
high-dimensional space. Hilbert spaces are spaces in which our geometry intuition from

A subset *convex* if for all

Let

Note that in non-Hilbert spaces, this may not be true! The proof is rather technical. See Young Chapter 3 or Moon and Stirling Chapter 2. Also known as the “closest point property”, this is very useful in compression and denoising.