What kind of topological vector space is $\mathbb{C}^n$?

Question

$\mathbb{R}^n$ is a n-dim Hilbert space with dot product as inner product. The topology induced by the inner product is what is used in real analysis.

I was wondering as what kind of topological vector space $\mathbb{C}^n$ is regarded, especially in complex analysis? Do you consider some inner product on it and therefore it may be a Hilbert space, or do you consider some norm on it and therefore it may be a Banach space?

Thanks and regards!

Hilbert space is by definition a Banach space. So it is both. — Asaf Karagila, May 07 '11 at 16:09
See Fabian's answer given here and follow the link he gives. Also, Qiaochu's answer here applies mutatis mutandis. — t.b., May 07 '11 at 16:11
@Asaf: I know that. I just wonder if $\mathbb{C}^n$ is studied as a Banach space only, or further as a Hilbert space, in complex analysis. How are its norm and inner product defined? — Tim, May 07 '11 at 16:12
Since you're an avid reader of Wikipedia, I really wonder what is unclear in this article on inner products? — t.b., May 07 '11 at 16:15
@Theo: Thanks! I am "flattered" for being recognized as "an avid reader of Wikipedia". — Tim, May 07 '11 at 16:20
Well, aren't many questions of yours (e.g. this one) inspired by your lecture of this great resource? — t.b., May 07 '11 at 16:25
I don't think my questions are duplicate of those linked in others' comment. I not just asked about the norm on $\mathbb{C}^n$, but also inner product on it. — Tim, May 07 '11 at 16:56
@Theo: How is Sylvester's law of inertia related to this discussion? — Tim, May 07 '11 at 18:22
Well, it tells you that there is essentially only one scalar product on $\mathbb{C}^n$. (I just see that the complex case isn't treated there, but it is rather straightforward). — t.b., May 07 '11 at 18:26

score 1 · Accepted Answer · answered May 07 '11 at 18:05

You will find the answer to your question for example in S. Krantz's review paper on several complex variables, which you can get from here. As he explains in the first pages, the standard inner product on $\mathbb{C}^n$ is $$\langle z,w \rangle = \sum_{k=1}^n z_k \bar{w_k},$$ and this gives the same topology as when identifying $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ in the usual way.

What kind of topological vector space is $\mathbb{C}^n$?

1 Answers1