# Azimuth Math Book Notes - Real Analysis, Serge Lang

• 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 $\langle w,v \rangle = \overline{\langle v,w \rangle}$.

Hermitian implies that $\langle v,v \rangle = \overline{\langle v,v \rangle}$, so $\langle v,v \rangle$ is real.

A hermitian form is called positive if $\langle v,v \rangle$ is never negative, and positive definite if it is always positive.

Orthogonal means that $\langle v,w \rangle = 0$.