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

$\mathbb{R}^n$ 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!

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Hilbert space is by definition a Banach space. So it is both. – Asaf Karagila May 7 '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 7 '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 7 '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 7 '11 at 16:15
@Theo: Thanks! I am "flattered" for being recognized as "an avid reader of Wikipedia". – Tim May 7 '11 at 16:20

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.