Inner Products and Norms

The Geometric Intuition

Imagine you are working in a space where vectors are not arrows in 3D space, but perhaps wavefunctions or abstract states. To make sense of these objects, we need a way to measure how "close" two states are or how "large" a state is. The inner product generalizes the familiar dot product from high school physics. In \(\mathbb{R}^n\), the dot product \(\mathbf{u} \cdot \mathbf{v}\) gives a scalar that depends on the magnitudes of the vectors and the angle between them. In an abstract vector space \(V\), we define a function \(\langle u, v \rangle\) that behaves similarly. It must be linear in one argument and conjugate-linear in the other (for complex spaces), and it must be positive-definite, meaning \(\langle v, v \rangle\) is always a non-negative real number, zero only if \(v\) is the zero vector. This positive-definiteness is crucial because it allows us to define a length, or norm, via \(\|v\| = \sqrt{\langle v, v \rangle}\). This turns the algebraic structure into a geometric one, enabling concepts like orthogonality (\(\langle u, v \rangle = 0\)) and projection.

Axioms of the Inner Product

Let \(V\) be a vector space over the field \(\mathbb{F}\) (where \(\mathbb{F}\) is either \(\mathbb{R}\) or \(\mathbb{C}\)). An inner product is a map \(\langle \cdot, \cdot \rangle : V \times V \to \mathbb{F}\) satisfying the following axioms for all \(u, v, w \in V\) and \(\alpha \in \mathbb{F}\):

1. Conjugate Symmetry: \(\langle u, v \rangle = \overline{\langle v, u \rangle}\). In the real case, this reduces to symmetry: \(\langle u, v \rangle = \langle v, u \rangle\).
2. Linearity in the First Argument: \(\langle \alpha u + v, w \rangle = \alpha \langle u, w \rangle + \langle v, w \rangle\). Note that in the complex case, the second argument is conjugate-linear: \(\langle u, \alpha v \rangle = \overline{\alpha} \langle u, v \rangle\).
3. Positive Definiteness: \(\langle v, v \rangle \ge 0\), and \(\langle v, v \rangle = 0\) if and only if \(v = 0\).

These axioms ensure that the inner product behaves consistently with our geometric intuition. For example, in \(\mathbb{C}^n\), the standard inner product is \(\langle u, v \rangle = \sum_{i=1}^n u_i \overline{v_i}\). The conjugation on the second argument is vital to ensure that \(\langle v, v \rangle = \sum |v_i|^2\) is real and non-negative.

The Induced Norm

Given an inner product, we define the norm (or length) of a vector \(v\) as:

This definition satisfies the three axioms of a norm: positivity, homogeneity, and the triangle inequality. Positivity and homogeneity follow directly from the inner product axioms. The triangle inequality, \(\|u + v\| \le \|u\| + \|v\|\), is a non-trivial result that relies on the Cauchy–Schwarz inequality. This inequality is the cornerstone of inner product spaces, linking the algebraic structure of the inner product to the geometric property of distance.

The norm allows us to define the distance between two vectors as \(d(u, v) = \|u - v\|\). This metric structure is essential for defining convergence, continuity, and completeness, which are prerequisites for Hilbert spaces in quantum mechanics.

Orthogonality and Orthonormal Bases

Two vectors \(u\) and \(v\) are said to be orthogonal if \(\langle u, v \rangle = 0\). A set of vectors \(\{e_1, e_2, \dots, e_n\}\) is orthonormal if \(\langle e_i, e_j \rangle = \delta_{ij}\), where \(\delta_{ij}\) is the Kronecker delta. Orthonormal bases are particularly useful because they simplify the representation of vectors. If \(\{e_i\}\) is an orthonormal basis, then any vector \(v\) can be written as \(v = \sum_{i=1}^n \langle v, e_i \rangle e_i\) (with the convention above, linear in the first argument; physicists write the same coefficient as \(\langle e_i | v \rangle\)). The coefficients \(\langle v, e_i \rangle\) are the projections of \(v\) onto the basis vectors.

In quantum mechanics, the basis vectors often represent eigenstates of an observable, and the coefficients are probability amplitudes. The norm squared of the coefficient, \(|\langle e_i, v \rangle|^2\), gives the probability of measuring the corresponding eigenvalue. This direct link between inner products and physical probabilities underscores the importance of this mathematical structure.