Finite vertex Hilbert space in the article convention

Let $V$ be a finite set. The complex vector space ${\mathbb{C}}^V$ consists of all functions $f:V\to \mathbb{C}$, with addition and scalar multiplication defined pointwise. Its vertex inner product, linear in the second argument, and its induced norm are

$⟨f,g⟩={\sum }_{x\in V}\overline{f(x)}g(x),\mathrm{\parallel }f{\mathrm{\parallel }}_2=\sqrt{⟨f,f⟩}\mathrm{.}$

Thus $f$ and $g$ are orthogonal when $⟨f,g⟩=0$. This is a complex Hilbert space: its induced norm metric is complete, meaning that every Cauchy sequence converges in ${\mathbb{C}}^V$.

A complex $V\times V$ matrix is an array $A=(A_{xy}{)}_{x,y\in V}$. It acts on ${\mathbb{C}}^V$ by $(Af)(x)={\sum }_{y\in V}{A}_{xy}f(y)$. Its conjugate transpose is the matrix $A^{\ast }$ with $(A^{\ast }{)}_{xy}=\overline{A_{yx}}$. An adjoint of this action is a linear map $T$ satisfying $⟨Af,g⟩=⟨f,Tg⟩$ for all $f,g$; here it is the action of $A^{\ast }$. The matrix is self-adjoint, or Hermitian, when $A=A^{\ast }$; in particular every real symmetric matrix, one satisfying $A_{xy}=A_{yx}\in \mathbb{R}$, is self-adjoint.

For comparison with the library's linear-first convention, put $(f,g{)}_{\mathrm{l}\mathrm{i}\mathrm{b}}={\sum }_{x\in V}f(x)\overline{g(x)}=\overline{⟨f,g⟩}$. The two conventions have the same norm and orthogonality. The adjoint and self-adjoint matrices are also the same in both conventions. For self-adjoint $A$, the quadratic values $⟨f,Af⟩$ and $(f,Af{)}_{\mathrm{l}\mathrm{i}\mathrm{b}}$ coincide and are real. Consequently a linear-first Cauchy–Schwarz bound transfers by taking absolute values, and a linear-first self-adjoint spectral statement transfers with the same operator, eigenvectors and real eigenvalues, subject to the hypotheses of the statement used.

Real and complex inner product spaces, with the inner product linear in the first argument

The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product

Finite rectangular matrices over a commutative ring, their entries, rows and columns

A finite sum in a commutative monoid indexed by an arbitrary finite set

Hilbert space

The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V

Linear map between vector spaces over the same field

Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis

Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism

Vector space over a field

Real and imaginary parts, complex conjugation, and modulus

Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive

The finite sums are independent of enumeration and include the empty sum (A finite sum in a commutative monoid indexed by an arbitrary finite set). Pointwise operations make ${\mathbb{C}}^V$ a complex vector space in the sense of Vector space over a field. Direct expansion using Real and imaginary parts, complex conjugation, and modulus and Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive shows that $(f,g{)}_{\mathrm{l}\mathrm{i}\mathrm{b}}$ is linear in $f$, conjugate symmetric and has diagonal ${\sum }_x\mathrm{\mid }f(x){\mathrm{\mid }}^2$, which is nonnegative and vanishes only when $f=0$; it is therefore an inner product in the library convention (Real and complex inner product spaces, with the inner product linear in the first argument). Its conjugate is the displayed second-linear pairing. Both have the same diagonal and hence the same induced norm (The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product), which is a norm by The induced length is a norm.

Choose an enumeration of the finite set $V$. The indicator functions of its singletons form an ordered basis in the sense of Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis: they span each function by its finite coordinate expansion, and evaluating a linear combination at each vertex proves independence. This includes the empty basis when $V=\mathrm{\varnothing }$. Thus Every finite-dimensional normed space is Banach makes the induced norm complete, and Hilbert space gives the Hilbert-space designation.

An enumeration identifies the array with a finite square matrix as in Finite rectangular matrices over a commutative ring, their entries, rows and columns; the displayed action is linear by distributivity, in the sense of Linear map between vector spaces over the same field. Interchanging the two finite sums gives $⟨Af,g⟩={\sum }_y\overline{f(y)}{\sum }_x\overline{A_{xy}}g(x)=⟨f,A^{\ast }g⟩$. The action of $A^{\ast }$ is linear, so it is an adjoint under the stated convention; it is unique because pairing the difference of two candidates with every $f$, including the difference itself, forces that difference to vanish. Conjugating the adjoint identity gives the defining linear-first identity of The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V, so the same adjoint works there.

Conjugation preserves zero and absolute value (Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive), proving the orthogonality and Cauchy–Schwarz transfers. If $A=A^{\ast }$, the adjoint identity and conjugate symmetry give $⟨f,Af⟩=\overline{⟨f,Af⟩}$; the value is real and hence equals its conjugate, the library-convention value. The eigenvector equation $Af=\lambda f$ with $f\mathrm{\ne }0$, as in Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism, does not involve a pairing, and self-adjointness is unchanged, so a spectral theorem in the library convention applies to the same operator and vectors whenever its own hypotheses hold.