Difference between $\mathbb{E}^n$ and $\mathbb{R}^n $?

Question

Basically what is the difference between $\mathbb{E}^n$ and $\mathbb{R}^n $? Simple question I know but I cannot think of what to type into google to get a proper answer.

Answer 1

$\mathbb {E}^n $ is a linear vector space of dimension $n $ for which there is a function, called inner product that is defined for each pair of vectors of $\mathbb {E}^n$ and that returns a scalar. When that inner product is defined we call the space an Euclidean Space. We know nothing about what the vectors actually are. We just know that the inner product exists and satisfies some axioms (in order to be an inner product the function must obey some rules). $\mathbb{R}^n$ is a specific example of an $n $-dimensional Euclidean Space where the vectors are what we know as those "arrows". And that is because we know how to define an inner product for $\mathbb {R}^n$.