What does "$S\times S\rightarrow R$

Question

In the textbook, Mathematical Methods and Algorighms for Signal Processing, Tood K. Moon, the $\mathbf{inner\;product}$ is defined it is a function $\langle\cdot,\cdot\rangle:S\times S\rightarrow R$ when $S$ is a vector space defined over a scalar field $R$. What does $S\times S\rightarrow R$ mean? I just know that inner product means like $x^Ty$ making some It is not intuitive to understand for me. Thank you.

Answer 1

It means that the inner product is simply a binary operation. Given any two elements of $S$ as input, the inner product will somehow convert the ordered pair into a new element that belongs in $R$ as output.

For example, take $S = \mathbb R^2$ and $R = \mathbb R$. Then using the dot product as our inner product, we have for example that: $$ \left\langle \underbrace{\begin{bmatrix} 1 \\ 3 \end{bmatrix}}_{\in ~ \mathbb R^2} , \underbrace{\begin{bmatrix} 4 \\ 2 \end{bmatrix}}_{\in ~ \mathbb R^2} \right\rangle = \underbrace{10}_{\in ~ \mathbb R} $$

Answer 2

The inner product that you know is an example of an inner product. It is a function which assigns a pair $(v,w)$ to an element in $R$, satisfying some axioms. The notation $S \times S \to R$ means in this case, that this is a function from the cartesian product (which consists of elements $(v,w) \in S \times S$ to the field $R$. But instead of writing $f: S \times S \to R$ and $f(v,w)$ we want to write $\langle v, w \rangle$.