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 EE be vector space over the complex numbers.

A sequilinear form is a function from E×EE \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)f(x + y) = f(x) + f(y) and f(cx)=c¯f(x)f(c x) = \overline{c} f(x).

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

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

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

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

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