Difference between Euclidean space and inner product space? Is it that Inner product space can have infinite dimensions?
 A: "A finite dimensional real inner product space is called a Euclidean space."
From Introduction to Hilbert Spaces with Applications, Second Edition by L. Debnath and P. Mikusinski, 1999.  Page 92.
A: The term Euclidian space is usually used only for spaces $\mathbb R^n$ for $n\in\mathbb N$. On the other hand, an inner product space is any vector space with a vector product.
A vector product induces a metric on the space, but that does not mean each inner product space is $\mathbb R^n$, as there exist inner product spaces which are not complete, for example. There also exist complete inner product spaces which are not finite-dimensional.
Bottom line: The difference is that Euclidian spaces are only one example of inner product spaces which have plenty of properties that inner product spaces in general do not.
A: The underlying field of a Euclidian space are the real numbers, $\mathbb R$. There are also complex spaces with inner products, like e.g. $\mathbb C^n$. The inner products on real spaces are bi-linear, while inner products on complex spaces are sesqui-linear, only.
A: There is a fine difference in the structure which you want to underline. I understand the Euclidean space to be $\mathbb{R}^n$ but considered as an affine space with a (euclidean) metric. You don't need to have a distinguished point such as zero.
