Difference between Euclidean space and inner product space?

Question

Is it that Inner product space can have infinite dimensions?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

"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.