Notes on Inner Product & Hilbert Space

## Basic Concepts

An **inner product** is a positive definite bilinear form (双线性形式) on a linear space:
for a linear space $X$ with underlying scalar field $\mathbb{F}$,
a map $\langle \cdot, \cdot \rangle: X \times X \to \mathbb{F}$ that satisfies

- Additivity: $\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$;
- Homogeneity: $\langle a x, y \rangle = a \langle x, y \rangle$, $\forall a \in \mathbb{F}$;
- Symmetry: $\langle y, x \rangle = \overline{\langle x, y \rangle}$;
- Positive definiteness: $\langle x, x \rangle > 0$, if $x \ne 0$;

An **inner product space** is a set with an inner product, $(X, \langle \cdot, \cdot \rangle)$.
Inner product specifies the *geometry* of a linear space.

Inner product space has norm $|x| = \sqrt{\langle x, x \rangle}$.

A **Hilbert Space** is a complete inner product space.

## Orthogonality

Two elements of an inner product space are **orthogonal** if their inner product is zero:
$\langle x, y \rangle = 0$.

Orthogonal complement

Orthogonal projection

Orthogonal set;
Orthogonal set;
Maximal/complete orthogonal set;
Orthonormal basis

Gram-Schmidt orthogonalization process

## Approximation and Fourier Series

For a linear space,
an **approximation** of a point on a (closed) linear subspace is
a point on the subspace that is closest to the point.

Theorem (Approximation in Banach and Hilbert spaces):
For a Banach space, an approximation may not exist.
For a Hilbert space, the approximation of any point on any linear subspace exists and is unique,
which is the orthogonal projection of that point.

Theorem (Riesz representation):

Theorem (Lax-Milgram):

🏷 Category=Analysis