Serge Lang, Real Analysis, Second Edition, Addison-Wesley, 1983. Text for first year graduate course in analysis.

Chapter 7, Hilbert Space

Let $E$ be vector space over the complex numbers.

A sequilinear form is a function from $E \times E$ into $\mathbb{C}$, which is linear in the first argument and semi-linear (“conjugate linear”) in the second argument, which means $f(x + y) = f(x) + f(y)$ and $f(c x) = \overline{c} f(x)$.

The form is written $\langle v, w \rangle$.

The form is called Hermitian if $\langle w,v \rangle = \overline{\langle v,w \rangle}$.

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