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[itex]\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[/itex]

[itex]\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[/itex]

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

[itex]\langle \psi | \phi \rangle = \int \psi^*(x) \phi(x) dx[/itex]

So it's the first argument, [itex]\psi[/itex] that is anti-linear.

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