Page 8 - Vector Analysis

4 CHAPTER 1. Linear Algebra

5. (αv, w) = α(v, w) for all α P F and v, w P V.

The bilinear form (¨, ¨) is called an inner product on V.

Example 1.14 (Standard Inner Product on Fn). Let F = R or C, and Fn be the vector
space defined in Example 1.3. A special inner product on the vector space Fn over F, called
the standard inner product on Fn, is defined by

n

(v, w) ” ÿ viwi ,

i=1

where vi and wi are the i-th component of v and w, respectively, and wi is the complex
conjugate of wi. We sometimes use v ¨ w to denote (v, w).

Example 1.15. Let V = C ([0, 1]; R). Define

ż1
(f, g) = f (x)g(x)dx .

0

()
Then C ([0, 1]; R), (¨, ¨) is an inner product space. The norm induced by this inner product

is given by [ż 1 ]1

}f } = |f (x)|2dx 2 ,

0

and is called the L2-norm.

Proposition 1.16. Let V be an inner product space with inner product (¨, ¨). The inner
product (¨, ¨) on V induces a norm defined by

}v} ” a(v, v)

satisfying

1. }v} ě 0 for all v P V.

2. }v} = 0 if and only if v = 0.

3. }αv} = |α|}v} for all α P F and v P V.

4. }v + w} ď }v} + }w} for all v, w P V.

5. |(v, w)| ď }v}}w} for all v, w P V.