The standard inner products make K n, ell two and quotient L two Hilbert spaces

In a real or complex inner-product space the pairing is linear in the first argument and conjugate-linear and conjugate-symmetric in the second, positive definite, and the induced length is the square root of the diagonal pairing (Real and complex inner-product spaces and their induced length).

For complex scalars $\mathrm{\mid }z{\mathrm{\mid }}^2=z\bar{z}\ge 0$ with $\mathrm{\mid }z\mathrm{\mid }=0$ exactly for $z=0$, and $\mathrm{\mid }zw\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive). The real field embeds in the complex field ($\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$); for an embedded real $t$, conjugation fixes $t$ and the complex modulus is $\sqrt{t^2}$, the real absolute value (Real and imaginary parts, complex conjugation, and modulus). Thus these same identities restrict to real scalars.

A normed space admitting a finite basis is a Banach space (Every finite-dimensional normed space is Banach), the induced length of an inner product is a norm (The induced length is a norm), and a Hilbert space is an inner-product space complete for its induced norm (Hilbert space).

On $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mathrm{\#})$ every scalar function is measurable, and the counting-measure dictionary gives $\int \mathrm{\mid }f{\mathrm{\mid }}^{2}d\mathrm{\#}={\sum }_k\mathrm{\mid }f(k){\mathrm{\mid }}^2$ (${\mathrm{\ell }}^p$ is the $L^p$ space of counting measure). Applying the same nonnegative identity to the positive and negative parts of the real and imaginary parts of an integrable complex function gives its absolutely convergent series as its integral, by the defining real/complex integral formulas (Integrable real and complex functions, and their integrals). Almost-everywhere equality is equality everywhere, since only the empty set has zero counting measure (${\mathrm{\ell }}^p$ is the $L^p$ space of counting measure, Counting measure on an arbitrary set).

On the quotient $L^2(\mu ;\mathbb{C})$ the pairing $⟨[f],[g]⟩=\int f\bar{g}$ is representative-independent and satisfies the inner-product axioms, with $⟨f,f⟩=\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$ (The complex $L^2$ pairing is well-defined and satisfies Cauchy–Schwarz).

Under countable choice, complex $L^p$ is complete and the complex $L^2$ pairing and its Cauchy–Schwarz inequality are available; the real $L^p$ spaces are complete for the same hypothesis (Complex completeness, density, and inner product: the consumer interface, Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$, The space $L^p(\mu )$ as the quotient by null functions).

Countable Choice is the hypothesis used by the cited $L^p$ completeness theorems; the complex $L^2$ pairing theorem [A5] itself is choice-free (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).