0.1 Inner Product & Inner Product Spaces
Given vector space \(\mathcal{V}\) an inner product \(\left< \cdot, \cdot \right>:\mathcal{V}\times\mathcal{V} \to \mathbb{R}\) is a mapping satisfying the following properties for all \(v_{1}, v_{2}, v_{3} \in \mathcal{V}\): 1. Symmetry: \(\left< v_{1}, v_{2} \right> = \left< v_{2}, v_{1} \right>\); 2. Non-negativity: \(\left< v_{1}, v_{1} \right> \geq 0\); 3. Non-Degeneracy: \(\left< v_{1}, v_{1} \right> =0 \iff v_{1}=0\); and 4. Linearity: \(\left< \alpha v_{1}, v_{2} + v_{3} \right> = \alpha \left< v_{1}, v_{2} \right> + \alpha \left< v_{1}, v_{3} \right> = \alpha \left< v_{1}, v_{2} + v_{3} \right>.\)
Note: Every inner product creates a norm satisfying: \(\lVert v \rVert = \sqrt{ \left< v, v \right> }.\)
A vector space with a corresponding inner product is known as an Inner Product Space. ### Inner Product Space Results
Some important inner-product space results include:
- The Cauchy-Schwartz Inequality: \(\lvert \left< v_{1}, v_{2} \right> \rvert\leq \lVert v_{1} \rVert \cdot \lVert v_{2} \rVert\).
- The Triangle Inequality: \(\lVert v_{1}+v_{2} \rVert \leq \lVert v_{1} \rVert + \lVert v_{2} \rVert\).