Finite-Dimensional Complex [[linear-algebra/hilbert-space|Hilbert Space]]

A finite-dimensional complex Hilbert space is a complex vector space $H$ of finite dimension equipped with an inner product

\langle \cdot,\cdot\rangle : H\times H \to \mathbb{C}

that is linear in one argument and conjugate-linear in the other (conventions vary), positive definite, and induces a norm $\|x\|=\sqrt{\langle x,x\rangle}$ . In finite dimension, every normed vector space is complete, so “Hilbert space” is automatic once the inner product is given.

This is the basic setting for finite-dimensional quantum mechanics: pure states are unit vectors up to a global phase, and more generally states are represented by Density Operator.

Equivalent concrete model

If $\dim H = n$ , then $H$ is (non-canonically) isomorphic to $\mathbb{C}^n$ with the standard inner product

\langle x,y\rangle = \sum_{k=1}^n \overline{x_k}\,y_k.

Choosing an orthonormal basis identifies vectors with column vectors and linear maps with matrices.

Orthonormal bases and expansions

An orthonormal basis $(e_1,\dots,e_n)$ satisfies $\langle e_i,e_j\rangle=\delta_{ij}$ . Every $x\in H$ has the expansion

x=\sum_{k=1}^n \langle e_k,x\rangle\,e_k, \qquad \|x\|^2=\sum_{k=1}^n |\langle e_k,x\rangle|^2.

The identity operator can be written as $I=\sum_{k=1}^n |e_k\rangle\langle e_k|$ in bra–ket notation.

Linear maps and adjoints

Every linear map $A:H\to H$ is automatically continuous and Bounded Operator Hilbert in finite dimension. The adjoint $A^\ast$ is the unique operator satisfying

\langle x,Ay\rangle=\langle A^\ast x,y\rangle \quad \text{for all } x,y\in H.

Self-adjoint operators (Self Adjoint Operator Observable) play the role of observables, with real eigenvalues and an orthonormal eigenbasis.

For the general (possibly infinite-dimensional) notion, see Hilbert Space.

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Finite-Dimensional Complex Hilbert Space