Inner product spaces are vector spaces with a special function that pairs vectors, giving us a way to measure angles and lengths. This function, called the inner product, follows specific rules that make it behave nicely with vector operations.

The inner product lets us define norms, which measure vector lengths. It also gives us the Cauchy-Schwarz inequality, a powerful tool for bounding inner products. These concepts are key to understanding vector spaces more deeply.

Inner Product Spaces

Definition of inner product spaces

• Vector space $V$ over a field $\mathbb{F}$ $(\mathbb{R}$ or $\mathbb{C})$ equipped with an inner product function $\langle \cdot, \cdot \rangle: V \times V \to \mathbb{F}$ that assigns a scalar value to each pair of vectors
• Inner product satisfies the following axioms for all $x, y, z \in V$ and $\alpha \in \mathbb{F}$:
  • Conjugate symmetry ensures $\langle x, y \rangle$ equals the complex conjugate of $\langle y, x \rangle$
  • Linearity in the second argument distributes the inner product over vector addition and scalar multiplication $\langle x, \alpha y + z \rangle = \alpha \langle x, y \rangle + \langle x, z \rangle$
  • Positive definiteness guarantees $\langle x, x \rangle \geq 0$ with equality if and only if $x = 0$, implying the inner product of a vector with itself is always non-negative and zero only for the zero vector

Properties of inner products

• Conjugate symmetry $\langle x, y \rangle = \overline{\langle y, x \rangle}$ holds for all $x, y \in V$ by the conjugate symmetry axiom
• Linearity in the first argument $\langle \alpha x + y, z \rangle = \alpha \langle x, z \rangle + \langle y, z \rangle$ for all $x, y, z \in V$ and $\alpha \in \mathbb{F}$ follows from conjugate symmetry and linearity in the second argument:
  • Proof: $\langle \alpha x + y, z \rangle = \overline{\langle z, \alpha x + y \rangle} = \overline{\alpha \langle z, x \rangle + \langle z, y \rangle} = \bar{\alpha} \overline{\langle z, x \rangle} + \overline{\langle z, y \rangle} = \alpha \langle x, z \rangle + \langle y, z \rangle$
• Sesquilinearity combines linearity in one argument and conjugate linearity in the other, resulting from linearity in both arguments and conjugate symmetry

Norms from inner products

• Inner product $\langle \cdot, \cdot \rangle$ on vector space $V$ induces a norm defined as $|x| = \sqrt{\langle x, x \rangle}$ for all $x \in V$, measuring the length or magnitude of a vector
• Induced norm satisfies properties:
  • Positivity $|x| \geq 0$ with $|x| = 0$ if and only if $x = 0$, ensuring non-negative lengths and zero length only for the zero vector
  • Homogeneity $|\alpha x| = |\alpha| |x|$ for all $\alpha \in \mathbb{F}$ and $x \in V$, scaling the length by the absolute value of the scalar
  • Triangle inequality $|x + y| \leq |x| + |y|$ for all $x, y \in V$, bounding the length of a sum of vectors by the sum of their lengths
• Cauchy-Schwarz inequality $|\langle x, y \rangle| \leq |x| |y|$ for all $x, y \in V$ geometrically interprets the absolute value of the inner product as bounded by the product of the vector norms

Identification of inner products

• Verifying a function $\langle \cdot, \cdot \rangle: V \times V \to \mathbb{F}$ is an inner product on vector space $V$ requires checking the inner product axioms:
  • Conjugate symmetry $\langle x, y \rangle = \overline{\langle y, x \rangle}$ for all $x, y \in V$
  • Linearity in the second argument $\langle x, \alpha y + z \rangle = \alpha \langle x, y \rangle + \langle x, z \rangle$ for all $x, y, z \in V$ and $\alpha \in \mathbb{F}$
  • Positive definiteness $\langle x, x \rangle \geq 0$ for all $x \in V$ with $\langle x, x \rangle = 0$ if and only if $x = 0$
• Common inner products include:
  • Euclidean inner product on $\mathbb{R}^n$: $\langle x, y \rangle = \sum_{i=1}^n x_i y_i$ for $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ (dot product)
  • Standard inner product on $\mathbb{C}^n$: $\langle x, y \rangle = \sum_{i=1}^n x_i \overline{y_i}$ for $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ (Hermitian inner product)
  • $L^2$ inner product on square-integrable functions: $\langle f, g \rangle = \int_a^b f(x) \overline{g(x)} dx$ for $f, g \in L^2([a, b])$ (integral of the product of one function with the complex conjugate of the other)