# Question About Hilbert Space Convention

Staff Emeritus
According to Wikipedia:http://en.wikipedia.org/wiki/Hilbert_space the inner product $\langle x | y \rangle$ is linear in the first argument and anti-linear in the second argument. That is:

$\langle \lambda_1 x_1 + \lambda_2 x_2 | y \rangle = \lambda_1 \langle x_1 | y \rangle + \lambda_2 \langle x_2 | y \rangle$

$\langle x | \lambda_1 y_1 + \lambda_2 y_2 \rangle = \lambda_1^* \langle x | y_1 \rangle + \lambda_2^* \langle x | y_2 \rangle$

That's just the opposite of what I always thought. I thought it was, for the usual Hilbert space of non-relativistic quantum mechanics:

$\langle \psi | \phi \rangle = \int \psi^*(x) \phi(x) dx$

So it's the first argument, $\psi$ that is anti-linear.

Is the quantum mechanics convention the opposite of the usual Hilbert-space convention, or am I confused?

Demystifier
Gold Member
You are right. Someone should correct the wikipedia.

martinbn
You're right in mathematics it is the opposite convention. I should actually say that in physics it is the opposite convention. The wiki article is correct, the physicists should be corrected.

vanhees71
Gold Member
2021 Award
Yes, this made me crazy when I took the lecture "Topological Vector Spaces" by a mathematician. Their convention spoils all the intuitive things due to the Dirac convention. For the exercises I used to calculate everything in terms of the physicist's convention and then translated to by just flipping the order of the scalar products, when it came to special case of the Hilbert space ;-).

• bhobba
dextercioby
Homework Helper
This is a well-known topic. I think wiki articles should follow the functional analysis convention ever since the first known book by MH Stone in 1932, then Riesz & Nagy 1954, etc.

bhobba
Mentor
Yes, this made me crazy when I took the lecture "Topological Vector Spaces" by a mathematician. Their convention spoils all the intuitive things due to the Dirac convention.

Aren't that the truth.

I learnt about Hilbert spaces in a math class - then had to unlearn it in physics      I simply learn't to live with it.

But I have to say the Dirac notation is addictive - even when mucking around with math stuff I use it - especially for Rigged Hilbert Spaces.

Thanks
Bill