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I have a simple question. Is the inner product $\langle x | z\rangle$ always real or does this hold only for an inner product with itself $\langle x | x \rangle$?

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$(x,z) \mapsto \overline{x} z$ (possibly conjugate, depending on how you like your inner product) is an inner product on $\mathbb{C}$. Hence it is not necessarily real. – copper.hat May 20 '13 at 14:49

2 Answers 2

In real vector spaces, both are real, and the latter is also positive.

In complex vector spaces, the first can be anything (e.g. $\left< x|(i | x\right>)$, but the latter is still real and positive.

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Hint: Any inner product $\langle {-} | {-} \rangle$ on a complex vector space satisfies $\langle \lambda x | y \rangle = \lambda \langle x | y \rangle$ for all $\lambda \in \mathbb{C}$.

You're right in saying that $\langle x|x \rangle$ is always real: in general, $\langle x|y \rangle = \overline{\langle y|x \rangle}$, so $\langle x|x \rangle = \overline{\langle x|x \rangle}$, so $\langle x|x \rangle$ is real. (It's also always positive.)

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I forgot to mention i need info for $\mathbb{C}$. – 71GA May 20 '13 at 14:43
What do you mean? – Clive Newstead May 20 '13 at 16:35

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