1 D. R. Wilton ECE Dept. ECE 6382 Introduction to Linear Vector Spaces Reference: D.G. Dudley, “Mathematical Foundations for Electromagnetic Theory,” IEEE Press, 1994.

Fields Fields

Linear Vector Spaces

Linear Vector Spaces, cont’d

Field Linear vector space A linear vector space enables us to form linear combinations of vector objects.

Linear Vector Space Examples

Linear Vector Space Examples, cont’d

Linear Independence

Dimensionality

Linear Independence and Dimensionality

Bases Note: If N is finite and dim S = N, then “and if” in the first line above may be replaced by “then”. I.e., any N independent vectors form a basis. Unfortunately, it is not the case that any infinite set of independent vectors forms a basis when dim S = ∞ !

Bases, cont’d

Inner Product Spaces Field Inner product space The inner product is a generalization of the dot product of vectors in R 3

Inner Product Spaces, cont’d

Since the inner product generalizes the notion of a dot product of vectors in R 3, we often read as “a dot b” and say that is a “projection of a along b ” or vice versa.

The Cauchy-Schwarz-Bunjakowsky (CSB) Inequality

The Cauchy-Schwarz-Bunjakowsky (CSB) Inequality, cont’d

Orthogonality and Orthonormality

Normed Linear Space

Normed Linear Space, cont’d

Convergence of a Sequence

Continuity of the Inner Product

Convergence in the Cauchy Sense

Convergence in the Cauchy Sense, cont’d

Hilbert Spaces

Hilbert Spaces, cont’d

Linear Subspaces

Linear Subspaces, cont’d

Gram-Schmidt Orthogonalization

Gram-Schmidt Orthogonalization, cont’d

Closed Sets

Best Approximation in a Hilbert Space

Best Approximation in a Hilbert Space, cont’d

Orthogonal Complement to a Linear Subspace

The Projection Theorem

The Projection Theorem and Best Approximation

The Projection Theorem and Best Approximation, cont’d

Operators in Hilbert Space

Operators in Hilbert Space, cont’d

Continuity of Hilbert Operators

Continuity of Hilbert Operators, cont’d

Equivalence of Boundedness and Continuity of Hilbert Operators

Unbounded Operator Example

Matrix Representation of Bounded Hilbert Operators

Matrix Representation of Bounded Hilbert Operators, cont’d

Non-Negative, Positive, and Positive Definite Operators

Non-Negative, Positive, and Positive Definite Operators, cont’d

The Moment Method

The Moment Method, cont’d