Inner product space, affine space, Cartesian space, and Euclidean space

Vector space, inner product space, Euclidean space … These terms appear frequently in the domains of reinforcement learning and optimization. In robotics, people usually make the distinction between “Cartesian space” and “joint space” for expressing coordinates. These are simple and fundamental concepts in algebra but can be confusing for some people. In this post, I give the formal definition of these technical terms from a roboticist’s perspective.

Vector space

A vector space over a field $F$ is a set $V$ together with two types of binary operations satisfying the eight axioms listed below. Elements of $V$ are called vectors. Elements of $F$ are called scalars

The two operations are:

• Vector addition $+$: $V\times V \rightarrow V$
• Scalar multiplication $\cdot$: $F\times V \rightarrow V$

The eight axioms are:

• Associativity of addition. $u + (v + w) = (u + v) + w$.
• Commutativity of addition. $u + v = v + u$.
• Identity element of addition. There exists an element $0\in V$, called the zero vector, such that $v + 0 = v$ for all $v \in V$.
• Inverse elements of addition. For every $v \in V$, there exists an element $-v \in V$, called the additive inverse of $v$, such that $v + (-v) = 0$.
• Compatibility of scalar multiplication with field multiplication. $a(bv) = (ab)v$.
• Identity element of scalar multiplication. $1v = v$, where $1$ denotes the multiplicative identity in $F$.
• Distributivity of scalar multiplication with respect to vector addition. $a(u + v) = au + av$.
• Distributivity of scalar multiplication with respect to field addition. $(a + b)v = av + bv$.

Inner product space

An inner product space (or pre-Hilbert space) is a vector space $V$ over the field $F$ ($F$ is either the field of real numbers $R$ or complex numbers $C$) equipped with an inner product, i.e. with a map $\langle \cdot,\cdot\rangle: V\times V\rightarrow F$ that satisfies the following properties for all vectors $x,y,z\in V$ and all scalars $a\in F$:

• Conjugate symmetry. $\langle x,y\rangle = \overline{\langle x,y\rangle}$.
• Linearity in the first argument. $\langle ax,y\rangle=a\langle x,y\rangle$ and $\langle x+y,z\rangle= \langle x,z\rangle+\langle y,z\rangle$.
• Positive semi-definite. $\langle x,x\rangle\geq 0$ for all $x\in V$.
• Non-degenerate $\langle x,y\rangle=0$ for all $y\in V$ is true if and only if $x=0$.

Affine space

An affine space is a set $A$ together with a vector space $V$, and an action of the vector space $V$ on the set $A$. The elements of an affine space are called points, and the elements of the associated vector space are called vectors or translations. The action is a mapping denoted as an addition

\begin{align} A\times V &\rightarrow A \\ (a,v) &\rightarrow a+v, \end{align}

that has the following three properties:

1. Right identity. $a+0=a$ for all $a\in A$, where $0$ is the zero vector in $V$.
2. Associativity. For all $v,w\in V$ and $a\in A$, we have $(a+v)+w=a+(v+w)$ (here the last $+$ is the addition in $V$).
3. Free and transitive action. For every $a\in A$, the mapping $V\rightarrow A: v\rightarrow a+v$ is a bijection.

In addition, from the property 1 and 2 above we can derive the “fourth property”:

• Existence of one-to-one translations. For all $v\in V$, the mapping $A\rightarrow A: a\rightarrow a+v$ is a bijection.

Property 3 has the following equivalent form (“fifth property”):

• Subtraction. For every $a,b\in A$, there exists a unique $v\in V$, denoted by $b-a$, such that $b=a+v$.

Intuitively, an affine space may be compared to “a vector space without origin or zero”. However, one should keep in mind that an affine space is not a vector space, but a set of points equipped with a vector space.

Euclidean space

A Euclidean vector space is a finite-dimensional inner product space $V$ over the field of real numbers $R$. A Euclidean space (NB: without the additional “vector” term) is an affine space over the reals such that the associated vector space is a Euclidean vector space. The dimension of a Euclidean space is the dimension of its associated vector space.

Cartesian space

A Cartesian space is a Euclidean space described by Cartesian coordinates, i.e., the coordinates of a point measured from an origin along a horizontal axis from left to right (the x-axis) and along a vertical axis from bottom to top (the y-axis). This can be extended to three-dimensions by the z-axis perpendicular to the x- and y-axes.