Chapter 10: Inner Product Spaces

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10.1 Inner Products and Norms

Definition 10.1 Inner Product Spaces

Example 10.1.1

Example 10.1.2

Example 10.1.3: 2

Theorem 10.1.1

Theorem 10.1.2

Remark

Example 10.1.4

Norm and Distance

Definition 10.2 Norm and Distance

Example 10.1.5

Example 10.1.6

Example 10.1.7

Theorem 10.1.3

Theorem 10.1.4: Cauchy-Schwarz Inequality4

Example 10.1.8

Theorem 10.1.5

Example 10.1.9

Theorem 10.1.6

Exercises for 10.1

Exercise 10.1.31

10.2 Orthogonal Sets of Vectors

Example 10.2.1

Theorem 10.2.1: Pythagoras’ Theorem

Theorem 10.2.2

Theorem 10.2.3

Example 10.2.2

Theorem 10.2.4: Expansion Theorem

Example 10.2.3

Lemma 10.2.1: Orthogonal Lemma

Theorem 10.2.5: Gram-Schmidt Orthogonalization Algorithm

Example 10.2.4

Corollary 10.2.1

Theorem 10.2.6

Definition 10.3 Orthogonal Projection on a Subspace

Theorem 10.2.7: Projection Theorem

Example 10.2.5

Theorem 10.2.8: Approximation Theorem

Example 10.2.6

Exercises for 10.2

Exercise 10.2.10

(Parseval’s formula).

Exercise 10.2.25

10.3 Orthogonal Diagonalization

Theorem 10.3.1

Definition 10.4 Diagonalizable Linear Operators

Example 10.3.1

Theorem 10.3.2

Example 10.3.2

Theorem 10.3.3

Example 10.3.3

Theorem 10.3.4

Theorem 10.3.5

Theorem 10.3.6: Principal Axes Theorem

Example 10.3.4

Exercises for 10.3

10.4 Isometries

Lemma 10.4.1

Definition 10.5 Isometries

Theorem 10.4.1

Theorem 10.4.2

Corollary 10.4.1

Example 10.4.1

Example 10.4.2

Theorem 10.4.3

Corollary 10.4.2

Example 10.4.3

Theorem 10.4.4

Corollary 10.4.3

Example 10.4.4

Lemma 10.4.2

Lemma 10.4.3

Lemma 10.4.4

Theorem 10.4.5

Theorem 10.4.6

Example 10.4.5

Theorem 10.4.7

Exercises for 10.4

10.5 An Application to Fourier Approximation6

Theorem 10.5.1

Example 10.5.1

Theorem 10.5.2

Example 10.5.2

Exercises for 10.5

Exercise 10.5.2

Exercise 10.5.3

Exercise 10.5.5

Section worksheets