The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$

Let $n\in \mathbb{N}$. A natural number is a von Neumann natural, that is a set, and $n=\{0,1,\dots ,n-1\}$ (The natural numbers $\mathbb{N}$ (von Neumann), On $\mathbb{N}$ the order is membership: $m<n⟺m\in n$), so

${\mathbb{R}}^n=\{x:x\text{ is a function }n\to \mathbb{R}\}$

is the function space of The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$ at $F=\mathbb{R}$ and $X=n$, a vector space over $\mathbb{R}$ under the pointwise operations (Vector space over a field). We write $x_k:=x(k)$ for $k<n$, and two elements of ${\mathbb{R}}^n$ are equal exactly when they agree at every $k<n$. This is the same set that ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it calls ${\mathbb{R}}^n$.

The Euclidean inner product of $x,y\in {\mathbb{R}}^n$ is the real number

$⟨x,y⟩:={\sum }_{k<n}{x}_k{y}_k,$

the finite sum of Finite sums and finite products, by recursion applied to the list $k\mapsto x_k{y}_k$ (extended by $0$ beyond $n$, as every finite list in this library is). The Euclidean norm of $x$ is

$\parallel x{\parallel }_2:=\sqrt{⟨x,x⟩},$

which is defined because $⟨x,x⟩={\sum }_{k<n}{x}_k^2\ge 0$ (a sum of nonnegative terms, Laws of finite sums and finite products clause 4 and Squares of nonzero elements are positive, the case $x_k=0$ giving $x_k^2=0$ by Integer powers $a^m$) and every nonnegative real has a unique nonnegative square root (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$).

Both are defined for every $n$, including $n=0$

At $n=0$ the set ${\mathbb{R}}^0$ has exactly one element, the empty function, and it is the zero vector space (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$ clause 5); the sum above is the empty sum, so $⟨x,y⟩=0$ and $\parallel x{\parallel }_2=0$. This is the first place on this page where the two index regimes diverge, and the divergence is deliberate. The published metrics $d_1$, $d_2$, $d_{\mathrm{\infty }}$ of ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it are defined only for $n\ge 1$, because $d_{\mathrm{\infty }}$ would otherwise be a maximum over the empty index set; the algebra above needs no such restriction. The boundary in this page runs between the algebra and the metric, not where a reader would guess, and Conventions of this page, the standing $n\ge 1$ hypothesis, and what is taken up elsewhere in the reading order lists exactly which items inherit $n\ge 1$.

The algebra of the inner product

For all $x,y,z\in {\mathbb{R}}^n$ and $\lambda \in \mathbb{R}$:

1. Symmetry. $⟨x,y⟩=⟨y,x⟩$, since $x_k{y}_k=y_k{x}_k$ termwise.
2. Additivity in the first argument. $⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩$: the list $k\mapsto (x_k+y_k)z_k$ is the termwise sum of $k\mapsto x_k{z}_k$ and $k\mapsto y_k{z}_k$, so Laws of finite sums and finite products clause 1 applies.
3. Homogeneity in the first argument. $⟨\lambda x,y⟩=\lambda ⟨x,y⟩$, by Laws of finite sums and finite products clause 2.
4. Bilinearity. Clauses 2 and 3 together with symmetry give the same two laws in the second argument.
5. Positive definiteness. $⟨x,x⟩\ge 0$, and $⟨x,x⟩=0$ if and only if $x=0$. Indeed a vanishing sum of nonnegative terms has every term $0$ (Laws of finite sums and finite products clause 4), so $x_k^2=0$ for every $k<n$, and a nonzero real has a positive square (Squares of nonzero elements are positive), whence $x_k=0$ for every $k<n$ and $x=0$.
6. Agreement with the published Euclidean metric. For $n\ge 1$ and $x,y\in {\mathbb{R}}^n$, $\parallel x-y{\parallel }_2=\sqrt{{\sum }_{k<n}(x_k-y_k{)}^2}=d_2(x,y)$, the two sides being the same expression (${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it). In particular $\parallel x{\parallel }_2=d_2(x,0)$.

That $\parallel \cdot {\parallel }_2$ is a norm in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms is proved in Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation, where the triangle inequality is obtained from the Cauchy-Schwarz inequality; it is not assumed here.

The natural numbers $\mathbb{N}$ (von Neumann)

On $\mathbb{N}$ the order is membership: $m<n⟺m\in n$

The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$

Vector space over a field

${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it

Finite sums and finite products, by recursion

Laws of finite sums and finite products

Squares of nonzero elements are positive

Integer powers $a^m$

Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$

The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$

Conventions of this page, the standing $n\ge 1$ hypothesis, and what is taken up elsewhere in the reading order

A norm on a real vector space, the induced metric, and the dictionary with the metric axioms

Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation

• Scope: the concrete form only. What is defined above is the Euclidean inner product on ${\mathbb{R}}^n$ and nothing more. The general theory of inner product spaces — abstract inner products, orthonormal bases, Gram-Schmidt, orthogonal projection and orthogonal complements of arbitrary subspaces — is developed elsewhere in this library. No item on this page claims anything about abstract inner product spaces, and no item on this page introduces the general notion.

• The standard basis and coordinates. For $i<n$ the standard unit vector $e_i\in {\mathbb{R}}^n$ has $e_i(i)=1$ and $e_i(j)=0$ for $j\mathrm{\ne }i$ (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$). Then $⟨x,e_i⟩=x_i$: the list $k\mapsto x_k(e_i{)}_k$ vanishes except at $k=i$, where its value is $x_i$, and a list vanishing off one index sums to its value there (Laws of finite sums and finite products clause 3, splitting the range at $i$). So the coordinates of $x$ are recovered by testing against the standard basis, which is the form used repeatedly below.

• Powers here are integer powers. $x_k^2$ means the integer power of Integer powers $a^m$, and $\parallel x{\parallel }_2^2=⟨x,x⟩$ by Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$ and Laws of integer exponents.