What does this notation mean in regards to the inner product?

Question

$$\langle \cdot , \cdot\rangle : \mathbb R^n\; \times \;\mathbb R^n \Rightarrow \mathbb R$$

Are the dots within the $\langle$ and $\rangle$ a pair of $n$ dimensional vectors that are mapping the dot product of the set $\mathbb R^n$ with itself which maps that set to the real numbers?

What it looks like to me is the vectors are mapping onto the Cartesian product of two $n$ dimensional spaces which are then mapped onto the real numbers.

Answer 1

A function is an ordered triple. Let's consider the function commonly written as $f(x) = 2 x + 1$. This function is actually three different sets: ( Domain, Range, Mapping ). For $f$, the function would be $f = \left( [-\infty,\infty], [-\infty,\infty], \{2x + 1 : x \in \mathbb{R}\}\right) = \left(\mathbb{R},\mathbb{R},\{2x+1:x\in\mathbb{R}\} \right).$

A way to say that something is a function is just to describe what the domain and range is $f:\text{Domain}\rightarrow\text{Range}$. In our case, $f:\mathbb{R}\rightarrow\mathbb{R}$. We sometimes write $f(\cdot)$ instead of $f$. If we were to do so, we would write $f(\cdot):\mathbb{R}\rightarrow\mathbb{R}$. And if we want to specify the mapping, we would do so afterwards. For our example we would say $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x)=2x+1$.

So what about your notation? The inner product is a function. It accepts two different vectors as input. With it's special notation of brackets, we denote this as $\langle\cdot,\cdot,\rangle$. What is its domain? The domain is the Cartesian cross product of the underlying field (which, in your case, looks like the real numbers): $\mathbb{R}\times\mathbb{R}$. And its range is the set of scalars.

So that leaves the notation: $\langle\cdot,\cdot\rangle:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$.

Answer 2

Yes the dots are a pair of vectors in $\mathbb{R}^n$. Inner products are a generalization of the familiar dot product of two vectors.

Answer 3

The "usual" notation for a function is to specify its "name", its domain and codomain, and (perhaps) a formula which assigns an element of the codomain to each element of the domain. For example $$ f : \mathbb{R} \to \mathbb{R} : x \mapsto x^2 $$ defines a function "named" $f$, which takes elements of $\mathbb{R}$ to $\mathbb{R}$ by squaring them.

This notation works well when the function is named by a variable, and we can use the usual prefix notation (e.g. the value that the sine function takes at a point $x$ is typically written by prefixing the name of the function to a value in the domain: $\sin x$). However, there are many functions in mathematics which don't use this kind of prefix notation: norms and absolute values are written with vertical bars which enclose the variable (e.g. $\|x\|_2$ might denote the $L^2$-norm of $x$), inner products are often written with angle braces (e.g. $\langle u, v \rangle$ denotes the inner product of $u$ and $v$), and so on.

For these kinds of infixed functions, it is nice to have a notation which makes it clear how the function behaves. The usual trick is to leave dots as placeholders. Thus $$ \langle \cdot , \cdot \rangle : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} $$ defines function which is "named" $\langle \cdot, \cdot \rangle$ and which takes elements of $\mathbb{R}^n\times \mathbb{R}^n$ to $\mathbb{R}$ via the (implicit) assignment $(x,y) \mapsto \langle x, y\rangle$. The dots just tell the reader where to plug in the variables (where, in this case, each of the variables is an $n$-dimensional vector with entries in $\mathbb{R}$).

This notation also comes up in places where a function may have multiple arguments, but only one of them is really important. For example, I work with functions of the kind $$ \zeta_E(\cdot, \delta) : \mathbb{C} \to \mathbb{C} : s \mapsto \int_{0}^{\delta} t^{s-1} |E| \,\mathrm{d}t. $$ Because $\delta$ is typically fixed early on (and therefore plays the role of a parameter), there is no good reason to specify where it comes from or what it does in the notation. However, it is still a variable on which the value of the function depends, so it is part of the notation. On the other hand, the complex variable $s$ is the thing that we are actually interested in varying, so its position is marked with a dot. The functions "name" is $\zeta_E(\cdot, \delta)$, whereas the value of the function at $s$ is denoted by $\zeta_E(s,\delta)$.