# Hilbert Spaces

Here, you find my whole video series about Hilbert spaces in the correct order and I also help you with some text around the videos. If you want to test your knowledge, please use the quizzes, and consult the PDF version of the video if needed. When you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:

#### Part 1 - Introductions and Cauchy-Schwarz Inequality

Let’s start with a short overview for the whole course and with the important definition of an inner product. This is something we have alreade discussed in Linear Algebra but now we also add an particular analysis part to it: we want to have completeness of the underlying normed space. In short, we have the following: a Hilbert space is an inner product space and a Banach space in one. We also use the first video here to prove the famous Cauchy–Bunyakovsky–Schwarz inequality.

#### Part 2 - Examples of Hilbert Spaces

Now we are ready to look at some examples for inner product spaces which are also complete. From the Functional Analysis series, we already know the important $\ell^2$-space. It consists of sequences which are also square-summable. It turns out that one can generalize this example to a so-called $L^2(\Omega, \mu)$-space. It consists of functions defined on a measure space $(\Omega, \mathcal{A}, \mu)$.