Hear ye, hear ye! Length and angles are a thing now!!

## Key Sequence

- We remembered how dot products work, then proceeded to generalize them into inner products—this is needed because complex numbers don’t behave well when squared so we need to add in special guardrails
- We then learned that dot products is just an instantiation of Euclidean Inner Product, which itself is simply one of many inner products. A vector space that has a well-defined inner product is now called an Inner Product Space
- Along with revisiting our definition of dot products to include complexes, we changed our definition of norm to be in terms of inner products \(\sqrt{\langle v,v \rangle}\) to help support complex vector spaces better; we then also redefined orthogonality and showed a few results regarding them
- Then, we did a bunch of analysis-y work to understand some properties of norms and inner products: Pythagorean Theorem, Cauchy-Schwartz Inequality, triangle inequality and parallelogram equality.

## New Definitions

- dot product and the more generally important…
- inner product!, Euclidean Inner Product, Inner Product Space
- norm
- orthogonal
- and now, a cornucopia of analysis

## Results and Their Proofs

- properties of the dot product
- properties of inner product
- properties of the norm
- orthogonality and \(0\)

## Questions for Jana

- How much of the analysis-y proof work do we have to remember for the analysis-y results?