Inner Product. The integral in the orthogonality condition

Chapter 10, Problem 35E

(choose chapter or problem)

Inner Product. The integral in the orthogonality condition (14) is like the dot product of two vectors in \(R^{n}\). In particular, show that the inner product of two functions defined by

(23)                        \(\langle f, g\rangle:=\int_{a}^{b} f(x) g(x) w(x) d x \)

where \(w(x)\) is a positive weight function, satisfies the following properties associated with the dot product (assume \(f, g \text { and } h\) are continuous on \({[a, b]}\)] :

(a) \(\langle f+g, h\rangle=\langle f, h\rangle+\langle g, h\rangle\).

(b) \(\langle c f, h\rangle=c\langle f, h\rangle\), where \(c\) is any real number.

(c) \(\langle f, g\rangle=\langle g, f\rangle\)

Equation Transcription:

Text Transcription:

R^n

\langle f, g\rangle:=\int_a^b f(x) g(x) w(x) d x

w(x)

f, g  and h

[a, b]

\langle f+g, h\rangle=\langle f, h\rangle+\langle g, h\rangle

\langle c f, h\rangle=c\langle f, h\rangle

c

\langle f, g\rangle=\langle g, f\rangle