Page 230 - A Course in Linear Algebra with Applications

214 Chapter Seven: Orthogonality in Vector Spaces

2
2
For brevity write a = ||v|| , b = < u, v > and c = ||u|| . Thus
2
at - 2bt + c =< u - tv, u - tv > > 0.
To see what this implies, complete the square in the usual
manner;

at2_ 2bt + c = a((t--) 2 + (--^)).
a a a 2 '
Since a > 0 and the expression on the left hand side of the
equation is non-negative for all values of t, it follows that
2
2
c/a > b /a , that is, b 2 < ac. On substituting the values of
a, b and c, and taking the square root, we obtain the desired
inequality.

Example 7.2.6
If 7.2.1 is applied to the vector space C[a, b] with the inner
product specified in Example 7.2.2, we obtain the inequality

2
2
1/2
f f(x)g(x)dx\ < ( f f(x) dx) 1/2 ( / g(x) dx) .
a J a J a
Normed linear spaces
The next step in our series of generalizations is to extend
the notion of length of a vector in Euclidean space. Let V
denote a real vector space. By a norm on V is meant a rule
which assigns to each vector v a real number ||v||, its norm,
such that the following properties hold:

(i) ||v|| > 0 and ||v|| = 0 if and only if v = 0;
(ii) ||cv|| = \c\ ||v||;
(iii) ||u +v|| < ||TU.|| + ||v||. (The Triangle Inequality).

These are to hold for all vectors u and v in V and all scalars
c. A vector space together with a norm is called a normed
linear space.