Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have just a quick question: what is the relationship/difference between a sesquilinear form $F(u,v)$ and a complex inner product $(u,v)$?

share|cite|improve this question
up vote 3 down vote accepted

A complex inner product $F$ is by definition a sesquilinear form, with some additional properties: we should have $F(v,v)\geq 0$ for each $v$ and $F(v,v)=0$ if and only if $v=0$.

You can construct in a vector space of dimension $2$ example of sesquilinear form such that we have $F(v,v)=0$ for all $v$ but $F(v_0,v_0)=0$ for some $v_0\neq 0$.

But a sesquilinear form doesn't need to be an inner product, for example $F(u,v)=0$ is obviously sesquilinear but not an inner product since we can have $F(v,v)= 0$ if $v\neq 0$ (except in the trivial case in which the vector space consists only of ${0}$).

share|cite|improve this answer
In your second sentence, did you mean to write $F(v,v) \neq 0$? – Rudy the Reindeer Aug 21 '12 at 10:15
Don't you need also that $\langle y,x\rangle=\overline{\langle x,y\rangle}$? – Alexander Frei Dec 21 '14 at 15:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.