Orthonormal Bases and Vector Spaces

April 10, 2026 · Luciano Muratore

Having explored the algebraic structures of groups, we now shift our focus toward the geometric and analytical structure of $\mathbb{R}^N$.

Vectors and Notation

A point in the $N$-dimensional real space $\mathbb{R}^{N}$ is specified by an $N$-tuple of coordinates. We denote this as a vector $\mathbf{x}$, typically expressed as a column vector:

$\mathbf{x} = \begin{bmatrix} x_{0} \\ x_{1} \\ \vdots \\ x_{N-1} \end{bmatrix} = [x_{0}, x_{1}, \dots, x_{N-1}]^{T}$

Each coordinate $x_i$ is a real number, and the $n$-th element specifically is denoted by $X_n$. In this framework, the zero vector $\mathbf{0}$ is simply the vector where every entry is zero.

Inner Product, Norm, and Distance

To discuss geometry, we need a way to measure angles and lengths. The inner product of two vectors $\mathbf{x}$ and $\mathbf{y}$ is defined as the sum of the products of their components:

$\langle \mathbf{x}, \mathbf{y} \rangle = \sum_{n=0}^{N-1} x_{n}y_{n}$

This leads us to two vital geometric concepts:

• Orthogonality: Two vectors are orthogonal ($\mathbf{x} \perp \mathbf{y}$) if and only if their inner product is zero.
• Norm: The $L_2$ norm, or magnitude, of a vector is given by $||\mathbf{x}||_{2} = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}$. [cite: 21, 24]

Furthermore, the distance between any two vectors is defined by the norm of their difference: $d(\mathbf{x}, \mathbf{y}) = ||\mathbf{x} - \mathbf{y}||_{2}$.

The Concept of a Basis

A set of $M$ vectors $\{x^{(k)}\}_{k=0}^{M-1}$ is said to span $\mathbb{R}^N$ if any vector $\mathbf{z} \in \mathbb{R}^N$ can be written as a linear combination of those vectors:

$\mathbf{z} = \sum_{k=0}^{M-1} \alpha_{k}x^{(k)}$

To successfully span the entirety of $\mathbb{R}^N$, a set must contain at least $N$ linearly independent vectors. When we have a set of exactly $N$ linearly independent vectors that spans the space, we call that set a basis.

Orthonormal Bases

A basis $\{y^{(k)}\}_{k=0}^{N-1}$ is particularly “well-behaved” if it is orthonormal. This means the vectors are both mutually orthogonal and have a unit norm, satisfying the condition:

$\langle y^{(k)}, y^{(h)} \rangle = \begin{cases} 1 & \text{if } k=h \\ 0 & \text{otherwise} \end{cases}$

The most common example is the canonical orthonormal basis. In this basis, each vector $\delta_{n}^{(k)}$ contains a single 1 at index $k$ and 0 at all other coordinates, providing the simplest possible coordinate system for the space.