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The normalized Hermitian form on a finite function space

Statement

Let $X$ be a nonempty finite set. On $C^{X}$ , with pointwise vector-space operations, define $⟨ f, h ⟩ = ∣ X ∣^{- 1} \sum_{x \in X} f (x) \overline{h (x)}$ . The denominator is the positive real image of $∣ X ∣$ . This is an inner product, linear in the first variable.

Facts & Assumptions

Given: $X$ finite and nonempty, $m = ∣ X ∣ > 0$ viewed in $R \subset C$ , and functions $f, h : X \to C$ .

[F1]

The linear-first inner-product axioms are linearity in the first variable, conjugate symmetry, and positive definiteness (Real and complex inner product spaces, with the inner product linear in the first argument).

[F2]

For $z = a + b i$ , $\overline{z} = a - b i$ and $∣ z ∣ = \sqrt{a^{2} + b^{2}}$ (Real and imaginary parts, complex conjugation, and modulus).

[F3]

Finite real sums are defined by enumeration, independently of that enumeration (The sum $\sum_{i \in S} a_{i}$ over a finite index set, and its product form).

[F4]

A finite sum of nonnegative reals is nonnegative and is zero only if every summand is zero (Laws of finite sums and finite products).

[F5]

Finite monoid sums are independent of enumeration and agree with real sums on real summands (A finite sum in a commutative monoid indexed by an arbitrary finite set).

[F6]

Conjugation preserves addition and multiplication, is involutive, and $z \overline{z} = ∣ z ∣^{2} \geq 0$ with equality exactly at $z = 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F7]

A property holding at zero and preserved under successor holds for every natural number (The principle of mathematical induction).

Proof

technique · direct

1.1

Pointwise addition and scaling make $C^{X}$ a vector space: each abelian addition identity, both distributive identities, scalar associativity and the scalar identity hold at each $x$ by the field laws in $C$ . F5 defines every complex sum in the displayed form. Since $m > 0$ , $m^{- 1}$ exists and is positive real, and F2 gives $\overline{m^{- 1}} = m^{- 1}$ . Thus the form is defined on every pair of functions.

F5F2givenalgebra

1.2

For complex lists $a_{j}, b_{j}$ and $c \in C$ , finite sums satisfy $\sum_{j} (a_{j} + b_{j}) = \sum_{j} a_{j} + \sum_{j} b_{j}$ , $\sum_{j} c a_{j} = c \sum_{j} a_{j}$ and $\overline{\sum_{j} a_{j}} = \sum_{j} \overline{a_{j}}$ . Here is the induction verifying their complex types: at length zero all sums vanish and $\overline{0} = 0$ . On appending $a, b$ , the first formula follows by rearranging $(A + B) + (a + b) = (A + a) + (B + b)$ ; the second from $c A + c a = c (A + a)$ ; the third from $\overline{A + a} = \overline{A} + \overline{a}$ . F7 proves the three identities for every length, and F5 transfers them to any finite enumeration of $X$ .

F5F6F7algebra

2.1

For $a, b \in C$ and $f_{1}, f_{2}, h \in C^{X}$ , expand the summand and apply step 1.2 to obtain $⟨ a f_{1} + b f_{2}, h ⟩ = m^{- 1} \sum_{x} (a f_{1} (x) + b f_{2} (x)) \overline{h (x)} = a ⟨ f_{1}, h ⟩ + b ⟨ f_{2}, h ⟩$ .

step 1.2algebra

2.2

Applying conjugation to the finite sum and using its involution gives $\overline{⟨ h, f ⟩} = m^{- 1} \sum_{x} \overline{h (x)} f (x) = ⟨ f, h ⟩$ .

step 1.1step 1.2F6algebra

2.3

On the diagonal, F6 gives $⟨ f, f ⟩ = m^{- 1} \sum_{x} ∣ f (x) ∣^{2}$ . These are nonnegative real summands, so F5 identifies their sum with the real finite sum of F3, and F4 shows the result is real and nonnegative. If it is zero, multiplication by $m > 0$ gives $\sum_{x} ∣ f (x) ∣^{2} = 0$ ; F4 forces each $∣ f (x) ∣^{2} = 0$ , hence $f (x) = 0$ by F6. Conversely $f = 0$ makes every summand zero.

F6F5F3F4step 1.1

3.1

Steps 2.1–2.3 verify F1 and therefore give an inner product. If $X = {x}$ the expression is $f (x) \overline{h (x)}$ and the same verification applies. The zero function has diagonal value zero by step 2.3; the empty set is excluded precisely because the prescribed normalization would divide by zero.

F1step 2.1step 2.2step 2.3∎

Sources

Axler, 6.2–6.3(a),(b), pp. 183–184, fixes the linear-first convention and positive weights. Etingof et al., §4.5 opening, p. 67, is the class-function specialization. The complex finite-sum laws and definiteness are explicitly derived above.

Depends on

Used by

The normalized Hermitian form on three points Example

Dependency tree · two levels

35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Etingof et al., Introduction to Representation Theory (standard reference, not scraped)
Sheldon Axler, Linear Algebra Done Right, fourth edition (standard reference, not scraped)