# Defintion of algebra of functions?

In the book "Information Geomery" by Nihat Ay et. al, Chapter 2.1 on page 25 starts as following:

We consider a non-empty and finite set $$I$$ [...] The real algebra of functions $$I \rightarrow \mathbb R$$ is denoted by $$\mathcal F(I)$$, and its unity $$\mathbb 1_{I}$$ or simply $$1$$ is given by $$\mathbb{1}(i) = 1$$, $$i\in I$$.

What exactly is an algebra of functions? I've searched the Internet, but found nothing yet. Any reference would be appreciated.

As you can tell by the suggested identity, it is simply the set of functions from $$I\to R$$ with pointwise multiplication as the multiplication operation. With the pointwise addition operation as addition, all the axioms for an algebra work out.

The notion you're looking for is that of an "algebra over a field." (Wikipedia link.)

Loosely, it is a special kind of vector space.

Let $$A$$ be a set and $$F$$ a field. Let us have the operations of addition ($$+ : A \times A \to A$$) and multiplication ($$\cdot : A \times A \to A$$). Note that this operation is distinct from that associated with scalar multiplication (a mapping $$A \times F \to A$$ normally denoted by just placing things next to each other).

$$A$$ (with these corresponding operations and their identities) is said to be an algebra over $$F$$ if the following hold true:

• $$A$$ is a vector space over $$F$$ (when considering addition and scalar multiplication)
• $$\cdot$$ distributes when multiplied on the left and right sides of a sum
• $$\cdot$$ is compatible with scalars, in the sense that $$\alpha x \cdot \beta y = (\alpha \beta)(x \cdot y)$$ for each $$\alpha,\beta \in F$$ and $$x,y \in A$$

The set $$\mathcal{F}(I)$$ of functions $$f : I \to \mathbb{R}$$ is an algebra over $$\mathbb{R}$$ if you define the operations correspondingly: with $$f,g \in \mathcal{F}(I)$$ and $$\alpha \in \mathbb{R}$$,

• Addition in $$\mathcal{F}(I)$$: $$(f+g)(x) := f(x) + g(x)$$
• Scalar Multiplication: $$(\alpha f)(x) := \alpha \cdot f(x)$$
• Multiplication in $$\mathcal{F}(I)$$: $$(f \cdot g)(x) := f(x) \cdot g(x)$$

(Here, we are playing a little fast and loose with the notation. Note that $$f+g$$ is addition in $$\mathcal{F}(I)$$, whereas $$f(x) + g(x)$$ is addition in $$\mathbb{R}$$. Similar ideas are used in the other bullets.)

You can prove from these definitions all of the axioms necessary to make $$\mathcal{F}(I)$$ a vector space and an algebra. (We simply call it an "algebra of functions" to emphasize that its elements are, in fact, functions -- but there's no other issue that arises from that, it's just an emphasis on what $$A$$ contains.)

The excerpt also specifies a function $$1_f$$ where $$1_f(x) = 1$$ for each $$x \in I$$. This is the multiplicative identity (or "unit") of $$\mathcal{F}(I)$$: it is the function whereby, for all $$g \in \mathcal{F}(I)$$,

$$(g \cdot 1_f)(x) = g(x) = (1_f \cdot g)(x)$$

(That is, multiplication by this function doesn't really "change" anything.)

This is not strictly necessary in the formulation of the definition of an algebra. However, like with rings, some people consider them by default to have a multiplicative identity, and sometimes not; algebras can suffer from the same definitional clashing. Less ambiguously, an algebra for which a unit exists is called a "unital algebra."

Of course, it might be useful to know that this algebra is unital for things mentioned later in the text.