Inner Product Spaces

An inner product space consists of a vector field \( V \) over the field \( F \) endowed with a function \( \langle . \ , \ . \rangle : V \times V \to F \) which satisfies the following properties: .

1. Conjugate Symmetry: \( \langle a, \ b \rangle = \overline{\langle b, \ a \rangle} \)
2. Linearity in the First Argument: \( \langle \alpha a, \ b \rangle = \alpha \langle a, \ b \rangle \) and \( \langle a + c, \ b \rangle = \langle a, \ b \rangle + \langle c, \ b \rangle \)
3. Positive-Definiteness: \( \langle a, \ a \rangle \ge 0 \), with equality holding iff \( a = \underline{0} \)

These properties imply many important properties such as the Triangle Inequality and the Cauchy Schwarz Inequality. The mapping is also an example of a sesquilinear form.

1. In \( \mathbb{R}^n \) we have \( \langle v_1, \ v_2 \rangle = a_1b_1 + a_2b_2 + ... + a_nb_n \)
2. For \( \mathbb{C}^n \) in order to preserve positive definiteness we take \( \langle v_1, \ v_2 \rangle = a_1\overline{b_1} + a_2\overline{b_2} + ... + a_n\overline{b_n} \)