I have been reading Gurtin, Introduction to Continuum Mechanics (1981) and on page 1, in the first sentence he writes:
The space under consideration will always be a three-dimensional
euclidean point space $\mathcal{E}$. The term point will be reserved for elements of $\mathcal{E}$, the term vector for elements of the associated vector space $\mathcal{V}$.
I have always thought that Euclidean space is a vector space (on real numbers with dot product). So what is meant by the statement from the book?
The other is a related question, what is the relationship between n dimensional (finite) Euclidean space and what is often denoted as $\mathbb{R}^n$? Is it that $\mathbb{R}^n$ is just one example of an Euclidean space (and they are isomorphic) but there are also other spaces which are Euclidean but not necessarily $\mathbb{R}^n$?
Best Answer
A Euclidean vector space is a pair $(E,\sigma)$ where $E$ is a finite-dimensional vector space over $\Bbb R$ and $\sigma\colon E\times E\rightarrow \Bbb R$ is a positive definite symmetric bilinear form.
We can identify the vector space $E$ with the coordinate space $\Bbb R^n$ after a choice of an orthonormal base, such that $\sigma(x,x)=\langle x,x\rangle$ is the usual scalar product.
A Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space.