Linear Algebra

Vector Spaces and Inner Products

The axioms, then an inner product. Carrying both over to function spaces is what most of the physics runs on.

Updated May 2026

A vector space is a set you can add and scale. That is nearly the whole definition, and it is deliberately thin: anything satisfying the axioms inherits every theorem proved from them, whether its elements are arrows, polynomials, or quantum states.

Inner products give you geometry

Addition and scaling alone give no notion of length or angle. An inner product does:

⟨ u, v ⟩ : V \times V \to C

conjugate-symmetric, linear in one argument, and positive definite. From it, $∥ v ∥ = ⟨ v, v ⟩$ , and orthogonality means $⟨ u, v ⟩ = 0$ .

Theorem

Cauchy–Schwarz

∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥

with equality precisely when $u$ and $v$ are parallel.

This one inequality is the triangle inequality, the correlation coefficient's bound of $\pm 1$ , and, applied to position and momentum operators, the Heisenberg uncertainty principle. Uncertainty is not a statement about disturbing what you measure; it is Cauchy–Schwarz applied to non-commuting operators.

Functions are vectors

Take continuous functions on $[a, b]$ with

⟨ f, g ⟩ = \int_{a}^{b} \overline{f (x)} g (x) w (x) d x

Every axiom holds. The space is infinite-dimensional, and the Sturm–Liouville eigenfunctions from the previous subject are an orthogonal basis for it.

Consequence

Fourier series, demystified

Expanding a function in sines and cosines is not a special technique. It is resolving a vector into components along an orthonormal basis:

c_{n} = \frac{⟨ e _{n} , f ⟩}{⟨ e _{n} , e _{n} ⟩}

the same formula as projecting an arrow onto an axis.

Completeness

Finite-dimensional spaces are all the same up to isomorphism. Infinite-dimensional ones are not, and the distinction that matters is whether Cauchy sequences converge inside the space.

A Hilbert space is an inner product space that is complete. $L^{2}$ , the square integrable functions, is the canonical example, and it is the state space of quantum mechanics. Completeness is what guarantees that an infinite superposition of states is still a state, which is not a technicality you can skip.

Gram–Schmidt

Any linearly independent set can be orthogonalised:

u_{k} = v_{k} - j < k \sum \frac{⟨ u _{j} , v _{k} ⟩}{⟨ u _{j} , u _{j} ⟩} u_{j}

Run it on ${1, x, x^{2}, \dots}$ over $[- 1, 1]$ and the Legendre polynomials come out. Run it with weight $e^{- x^{2}}$ over the real line and you get Hermite polynomials. The classical orthogonal polynomials are Gram–Schmidt applied to monomials under different weights, nothing more exotic than that.

Dual spaces, and why physicists write bras

Every vector space has a dual $V^{*}$ : the space of linear maps $V \to C$ . In finite dimensions the two have the same dimension, and an inner product gives a canonical correspondence between them.

That correspondence is exactly Dirac notation. The ket $∣ ψ ⟩$ is a vector; the bra $⟨ ψ ∣$ is the element of the dual space the inner product associates with it; and $⟨ ϕ ∣ ψ ⟩$ is that functional applied to that vector. The notation is not a physicist's shorthand for something sloppier. It is the cleanest available way of writing the pairing between a space and its dual.

Theorem

Riesz representation

In a Hilbert space, every continuous linear functional $f$ is given by an inner product with a unique vector $v_{f}$ :

f (x) = ⟨ v_{f}, x ⟩

This is what guarantees the bra–ket correspondence is well defined in infinite dimensions, where "same dimension" is no longer an argument. It is also the reason observables can be represented by operators at all.