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In https://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.2012-08-13
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    You mean there's something wrong on the internet? Noooooooooo!2012-08-13
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    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**.2012-08-13
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    @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.2012-08-13
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    Just saying pre-Hilbert space doesn't specify whether it's complex or not though.2012-08-13
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    @PeteL.Clark sorry to drag this up, I'm trying to work out if my book has a typo for $(x,x)\ge 0$ - it has $(x,x)=0\iff x=0$, I guessed that "unitary space" was an archaic (old book) term for inner product but I am convinced that the requirement $(x|y)\ge 0$ is a typo, especially as $(x|y)=\overline{(y|x)}$ (what would $\ge$ mean her?) can you confirm? Also is the notation $(x|y)$ trivially translatable to $\langle x,y\rangle$? (It seems to be - I want a second opinion)2015-11-24