Inner Product Space
1 Inner Product
Given linear 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 \(v_{1}, v_{2}, v_{3} \in \mathcal{V}\):
- Symmetry: \(\left< v_{1}, v_{2} \right> = \left< v_{2}, v_{1} \right>\);
- Non-negativity: \(\left< v_{1}, v_{1} \right> \geq 0\);
- Non-Degeneracy: \(\left< v_{1}, v_{1} \right> =0 \iff v_{1}=0\); and
- 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: \(\lVert v \rVert = \sqrt{ \left< v, v \right> }.\)
2 Inner Product Space
A vector space with a corresponding inner product is known as an Inner Product Space.
An inner product space is said to be complete if every Cauchy Sequence \(\{v_{n}\}\) in the inner product space, there exists a limit \(v \in \mathcal{V}\) such that \(\lim_{n \to \infty} v_{n} = v\).
A complete inner product space is known as a Hilbert Space.
2.1 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]]: \(\lVert v_{1}+v_{2} \rVert \leq \lVert v_{1} \rVert + \lVert v_{2} \rVert\).