Complex vs. Real Inner-Product Spaces

Question

I'm studying some unitary and orthogonal operators, and from what I understand, $T$, a linear operator on a finite-dimensional inner-product space $V$ over $F$, is unitary/orthogonal if the norm of $T(x)$ is the same as the norm of $x$, AND $F = \mathbb{C}$ or $F = \mathbb{R}$, respectively.

I think I'm confusing myself on complex and real inner-product spaces; a complex inner-product space is simply an inner-product space that allows for the existence of complex numbers, right? So, every number is of the form $a + b\mathrm{i}$. Real inner-product spaces, however, means $b = 0$ for all numbers $a + b\mathrm{i}$, right (essentially, imaginary numbers cannot exist)? Is this true, false, or more to it? Thank you for your help.

Answer 1

A real inner product space is a vector space over the field $ \mathbb R$ together with an inner product.

A complex inner product space is a vector space over the field $ \mathbb C$ together with an inner product.