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):