6

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?

hardmath
  • 37,015
Sunni
  • 4,536

1 Answers1

16

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

kahen
  • 15,760
  • 3
    But not necessarily complete, so it's also definitely not the same as complex Hilbert space. – tomasz Aug 13 '12 at 13:55
  • 5
    You mean there's something wrong on the internet? Noooooooooo! – Pete L. Clark Aug 13 '12 at 13:59
  • 1
    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
  • 1
    @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
  • @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) – Alec Teal Nov 24 '15 at 22:15