Say I have a complex-valued vector space $\mathbb{C}^n$ with the following inner product:
$$
\langle u,v\rangle=u^Tv
$$
If one picks only vectors that have real entries, and only allow linear transformations from $\mathbb{R}^n\to \mathbb{R}^n$.
Then, can we say that this complex vector space embeds a real Hilbert sub-space, in some sense?
Best Answer
The definition of a vector space includes the field over which the elements of the vector space are included. Since you’re saying you’re vector space is $\mathbb{C}^N$, you’re implicitly saying that your vector space is over the complex values. A property of a vector space is that a vector from the space multiplied by any scalar from the field is also in the space. Therefore, if you give me a real vector, I must be able to multiply it by i and that vector is also in the space.
So this contradicts the idea that you can only choose real vectors. Once you give me a real vector, I can make complex vectors that lie in the space.
Note that if you had defined your inner product to be $Re\{u^* v\}$, then you would you have an inner product space that is equivalent to the inner product space of $\mathbb{R}^{2N}$ and the regular dot product. So that is a Hilbert space.