Index
applications, 75
bijective, 17
linear regression, 75
codomain, 16
Wiener-Hopf, 77
convex, 49
domain, 16
Banach space
global minimum value, 48
best approximation, 65
image, 17
closed subspace, 47
injective, 16
Schauder basis, 47
inverse function, 17
standard Schauder basis, 47
inverse image, 17
strictly convex, 48
local minimum value, 48
one-to-one, 16
eigenvalues, 113
algebraic multiplicity, 114
one-to-one correspondence, 17
characteristic polynomial, 114
onto, 17
defective, 119
preimage, 17
diagonalizable, 115
surjective, 17
eigenspace, 114
eigenvalue, 113
inner-product space, 50
adjoint, 96
eigenvector, 113
best approximation, 65
generalized eigenspace, 120
Cauchy-Schwarz inequality, 53
generalized eigenvector, 121
dual approximation, 80
geometric multiplicity, 114
Euclidean space, 52
Jordan chain, 121
Gram-Schmidt orthogonalization, 56
Jordan normal form, 120
Hilbert space, 57
similar, 116
induced norm, 52
spectrum, 113
inner product, 49
field, 35
least-squares, 72
functions, 16
normal equations, 70
135
136
INDEX
orthogonal, 54
contrapositive, 9
orthogonal complement, 56
converse, 7
orthogonal set, 54
corollary, 10
orthonormal basis, 57
disjunction, 3
Parseval identity, 58
existential quantifier, 9
Riesz representation theorem, 94
fallacy, 9
standard inner product, 50
implication relation, 4
integral, 46
lemma, 10
almost everywhere, 46
logical equivalence, 8
Lebesgue integral, 46
logical implication, 6
Riemann integral, 46
negation, 3
proof, 1
linear transform, 59
proof by contradiction, 9
bounded, 90
proposition, 10
idempotent, 62
tautology, 5
invertible, 85
theorem, 10
linear operator, 84
universal quantifier, 9
non-singular, 59
nullity, 61
matrix
nullspace, 61
compact SVD, 129
operator norm, 89
convergent, 122
orthogonal projection, 63
Frobenius norm, 94
pseudoinverse, 101
Grammian, 70
range, 60
Hermitian transpose, 38
rank, 61
inverse, 37
singular, 59
matrix product, 37
transpose, 89
orthogonal, 112
logic, 1
positive-definite, 71
biconditional, 4
positive-semidefinite, 71
conditional, 3
projection matrix, 74
conditional connective, 4
pseudoinverse, 74, 101
conjecture, 1
spectral radius, 93
conjunction, 2
trace, 86
contradiction, 5
unitary, 112
INDEX
matrix factorization, 103
137
equivalence classes, 15
backward substitution, 104
equivalence relation, 15
Cholesky factorization, 110
integers, 12
forward substitution, 103
intersection, 14
LDLT decomposition, 110
naive set theory, 12
lower triangular, 103
natural numbers, 13
LU decomposition, 105
partial order, 134
orthogonal, 111
quotient set, 15
unit triangular, 103
rational numbers, 13
unitary, 111
real numbers, 12
upper triangular, 103
Russell’s Paradox, 13
metric space, 19
set, 12
d-continuous, 22
set difference, 14
d-converges, 21
set-builder notation, 13
d-open, 21
simple order <, 132
d-open ball, 20
singleton, 12
Cauchy sequence, 21
strict partial order, 133
complete, 22
subset, 13
contraction, 29
union, 14
distance, 20
well-ordered, 132
Euclidean metric, 20
isometry, 23
topology, 23
metric, 19
basis, 23
points, 20
closed, 24
sequence, 20
closure, 25
compact, 31
set theory
complete, 28
axiom of choice, 131
completion, 29
Cartesian Product, 15
continuous, 26
complement, 14
converge, 27
complex numbers, 12
dense, 25
disjoint, 14
extended real numbers, 31
elements, 12
infimum, 32
empty set, 13
interior, 25
138
INDEX
limit point, 25
maximum, 31
metric topology, 24
metrizable, 24
minimum, 32
neighborhood, 25
open set, 23
separable, 25
supremum, 31
totally bounded, 30
vector space, 37
Banach space, 47
convex set, 48
dimension, 42
direct sum, 39
dual basis, 88
dual space, 87
finite-dimensional, 40
functional, 48
Hamel basis, 40
homomorphism, 86
isomorphism, 86
linear combination, 38
linear functional, 86
linear transform, 59
linearly dependent, 40
linearly independent, 40
norm, 45
normalized, 46
ordered basis, 43
span, 39
standard basis, 41
subspace, 39
unit vector, 46