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In http://www.encyclopediaofmath.org/index.php/Unitary_space, unitary space seems to be Hilbert space. But in http://www.answers.com/topic/unitary-space, "finite dimensional" is required. My question is, which definition of unitary space is commonly used?

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Answers.com is wrong. Unitary space is an archaic name for complex inner product space.

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But not necessarily complete, so it's also definitely not the same as complex Hilbert space. – tomasz Aug 13 '12 at 13:55
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You mean there's something wrong on the internet? Noooooooooo! – Pete L. Clark Aug 13 '12 at 13:59
Seriously, as kahen says, the term "unitary space" is not itself commonly used nowadays, so far as I'm aware. Instead people (especially analysts) will speak of complex inner product spaces while others (especially algebraists) will speak of Hermitian spaces. – Pete L. Clark Aug 13 '12 at 14:01
@PeteL.Clark: To be fair, I remember the term used in my first year linear algebra course (it did focus mostly on finite-dimensional spaces, and I can't recall if the definition used there assumed finite dimension, but I doubt it did). Also, another term I've heard is pre-Hilbert space. – tomasz Aug 13 '12 at 14:05
Just saying pre-Hilbert space doesn't specify whether it's complex or not though. – kahen Aug 13 '12 at 14:39

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