What is an extension of an inner product space?

Question

Chapter 5 of Alan Macdonald's "Linear and Geometric Algebra" begins by saying,

The geometric algebra $\mathbb{G}^n $ is an extension of the inner product space $\mathbb{R}^n.$

What is an extension of an inner product space?

Answer 1

According to dictionary.com, "extension" can mean "an instance of enlarging the scope of something". And according to Wikipedia, an "inner product space" is "a vector space with an additional structure called an inner product" and while this is alluded to early on, later it clearly states "the real n-space ${\displaystyle \mathbb {R} ^{n}}$ with the dot product is an inner product space".

Putting this together, an "extension of the inner product space $\mathbb R^n$" means taking the vectors and inner product (standardly the dot product) of $\mathbb R^n$, and somehow enlarging the scope. That's a little vague, but the default interpretation would be that, at minimum: $\mathbb G^n$ is a superset of $\mathbb R^n$, and you can still take the inner (dot) product of the $\mathbb R^n$ vectors inside of $\mathbb G^n$. (The previous chapter on $\mathbb G^3$ and the later text in this chapter clarifies what $\mathbb G^n$ is more precisely than this one-sentence summary.)

Answer 2

I mean extension in the usual English sense: $\mathbb{R}$, its vectors and its operations (scalar product, vector addition, inner product) are contained in $\mathbb{G}$.