MATH 323 Lecture 22

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Inner Product and Norm

Cauchy-Schwarz Inequality

Can be rewritten as

is normed linear space if with each , the number \|v\| is associated:

  1. and
  2. (triangle inequality)

Theorem 5.4.3

If is an inner product space, then is a norm.

Euclidean Norm

, , is a norm

Euclidean Norm given by is defined for as

Supreme norm



Applications in Orthogonality

Inner product space ,

For vectors such that when , is an orthogonal set of vectors.

If we transform into a set of unit vectors , then

This is called an orthonormal set

Theorem 5.5.1

If is an orthogonal set of nonzero vectors, then the vectors are linearly independent.

Theorem 5.5.2

An orthonormal set is a basis for and is called an orthonormal basis

Let be an orthonormal basis for .


For an orthonormal basis ,

Corollary: Parseval's Formula

For an orthonormal basis ,