Inner product space

Dependencies:

1. Field
2. Vector Space

Let $F$ be a subfield of $\mathbb{C}$ (complex numbers). Let $V$ be a vector space over $F$.

The inner product is a function from $V \times V$ to $F$. The inner product of $\mathbf{u}$ and $\mathbf{v}$ is denoted as $\langle \mathbf{u}, \mathbf{v} \rangle$.

$V$ is an inner product space if it satisfies the following properties:

• Conjugate symmetry: $\langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle}$.
• Linearity in the first argument:
  • $\langle \mathbf{u_1} + \mathbf{u_2}, \mathbf{v} \rangle = \langle \mathbf{u_1}, \mathbf{v} \rangle + \langle \mathbf{u_2}, \mathbf{v} \rangle$.
  • $a\langle \mathbf{u}, \mathbf{v} \rangle = \langle a\mathbf{u}, \mathbf{v} \rangle$.
• Positive definiteness:
  • (positive semidefiniteness) $\langle \mathbf{v}, \mathbf{v} \rangle \ge 0$.
  • $\langle \mathbf{v}, \mathbf{v} \rangle = 0 \iff \mathbf{v} = 0$.

$\langle \mathbf{v}, \mathbf{v} \rangle$ is also denoted as $\|\mathbf{v}\|^2$. Also, $\operatorname{norm}(\mathbf{v}) = \|\mathbf{v}\| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle}$.

Dependency for:

1. Cauchy-Schwarz inequality for random variables
2. Extreme direction of convex cone as extreme point of intersection with hyperplane
3. Vertex of a set
4. Vertex implies extreme point
5. Symmetric operator
6. Triangle inequality
7. x and y are parallel iff ∥x∥²∥y∥² = |< x, y >|².
8. Coordinatization over orthogonal vectors
9. Pythagorean theorem
10. Zero in inner product
11. Cauchy-Schwarz Inequality
12. A set of mutually orthogonal vectors is linearly independent
13. Gram-Schmidt Process
14. Orthogonality and orthonormality
15. Joining orthogonal linindep sets
16. Inner product is anti-linear in second argument
17. Orthonormal basis change matrix
18. Matrices form an inner-product space

Info:

• Depth: 4
• Number of transitive dependencies: 4

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Vector Space