A

Abelian group, 40
abstract algebra, 33
additive group, 46
adherent point, 275
adjoint system of
  ordinary differential
  equations, 261
adjoint transformation, 219, 220,422
affine linear subspace, 85
algebra, 30, 56, 57,104
algebraically closed
  field, 165
algebraic conjugate, 110
algebraic multiplicity, 167,223
algebraic structure, 31
algebraic system, 30
algebra with identity, 57,105
aligned, 379
almost everywhere, 295
approximate eigenvalue, 444
approximate point
  spectrum, 444
approximation, 395
Arzela-Ascoli theorem, 316
Ascoli's lemma, 317
associative algebra, 56,105
associative operation, 28
automorphism, 64, 68
autonomous system of
  differential equations, 241
Axioms of norm, 207

B

Banach inverse theorem, 416
Banach space, 31, 345
basis, 61, 89
Bessel inequality, 213, 380
bicompact, 302
bijection 14
bijective, 14, 100
bilinear form, 114
bilinear functional, 114-115
binary operation, 26
block diagonal matrix, 175
Bolzano-Weierstrass
  property, 302
Bolzano-Weierstrass
  theorem, 298
boundary, 279
bounded linear
  functional, 356
bounded linear operator, 407
bounded metric space, 265
bounded sequence, 286
B(X,Y),409

C

C[a, b] ,80
cancellation laws, 34
canonical mapping, 372
cardinal number, 24
cartesian product, 10
Cauchy-Peano
  existence theorem, 332
Cauchy sequence, 290
Cayley-Hamilton theorem, 167
Cayley's theorem, 66
characteristic equation, 166, 259
characteristic polynomial, 166
characteristic value, 164
characteristic vector, 164
C^∞, 79
classical adjoint of a
  matrix, 162
closed interval, 283
closed relative to an
  operation, 28
closed set, 279
closed sphere, 283
closure, 275
Cⁿ,78
cofactor, 158
colinear, 379
collection of su bsets, 8
column matriX, 132
column of a matriX, 132
column rank of a matrix, 152
column vector, 125
commutative algebra, 57, IDS
commutative group, 40
commutative operation, 28
commutative ring, 47
compact, 302
compact operator, 447
companion form, 256
comparable matrices, 137
complement of a subset, 4
completely continuous
  operator, 447
complete metric space, 290
complete ortghonormal
  set of vectors, 213, 389
completion, 295
complex vector space, 76
composite function, 16
composite mathematical
  system, 30, 54
conformal matrices, 137
congruent matrices, 198
conjugate functional, 114
conjugate operator, 421
constant coefficients, 241
contact point, 275
continuation of a
  solution, 336
continuous function, 307,408
continuous spectrum, 440
contraction mapping, 314
converge, 286, 350
convex, 351-355
coordinate representation
  of a vector, 125
coordinates of a vector
  with respect to a basis, 92, 124
countable set, 23
countably infinite set, 23
covering, 299
cyclic group, 43,44

D

degree of a polynomial, 70
DeMorgan's laws, 7, 12
dense-in-itself, 284
denumerable set, 23
derived set, 277-278
determinant of a
  linear transformation, 163
determinant of a matrix, 157
diagonalization of a
  matrix, 172
diagonalization process, 450
diagonal matrix, 155
diameter of a set, 267
difference of sets, 7
differentiation:
  of matrices, 247
  of vectors, 241
dimension, 78, 92, 392
direct product, 10
direct sum of linear,
  subspaces 83,457
discrete metric, 265
disjoint sets,S
disjoint vector spaces, 83
distance 264
  between a point
     and a set, 267
  between sets, 267
  between vectors, 208
distribution function 397
distributive, 28 '
diverge, 286,350
division algorithm, 71
division (of
  polynomials), 72
division ring, 46, 50
divisor, 49
divisors of zero, 48
domain of a function, 12
domain of a relation, 25
dot product, 114
dual, 358
dual basis, 112

E

ε-approxirnate solution, 329
ε-dense set, 299
ε-net,299
eigenvalue, 164,439
eigenvector, 164,439
element, 2
element of ordered set, 10
empty set, 3
endomorphism, 64, 68
equal by definition, 10
equality of functions, 14
equality of matrices, 132
equality of sets, 3
equals relation, 26
equieontinuous, 316
equivalence relation, 26
equivalent matrices, 151
equivalent metrics, 318
equivalent sets, 23
error vector, 395
estimate, 398
Euclidean metric, 271
Euclidean norm, 207
Euclidean space, 30,124,205
even permutation, 156
events, 397
everywhere dense, 284
expected value, 398
extended real line, 266
extended real numbers, 266
extension of a function, 20
extension of an
  operation, 29
exterior, 279
extremum, 464

F

factor, 72
family of disjoint sets, 12
family of subsets, 8
field, 30, 46, 50
  field of complex
numbers, 51
field of real numbers, 51
finite covering, 299
finite-dimensional
  operator, 450
finite-dimensional
  vector space, 92, 124
finite group, 40
  finite intersection
property, 305
finite linear
  combination of vectors, 85
finite set, 8
fixed point, 3 15
flat, 85
Fⁿ,78
Fourier coefficients, 380, 389
Frechel derivative, 458
Fredholm equation, 97,326
Fredholm operator, 425
function, 12
functional, 109,355
functional analysis, 343
function space, 80
fundamental matrix, 246
fundamental sequence, 290
fundamental set, 246
fundamental theorem
  of algebra, 74
fundamental theorem of
  linear equations, 99

G

Gateaux differential, 458
generalized associative
  law,36
generated subspace, 383
generators of a set, 60
Gram matrix, 395
Gram-Schmidt process, 213, 391
graph of a function, 14
greatest common divisor, 73
Gronwall inequality, 332
group, 30, 39
group component, 46
group operation, 46

H

Hahn-Banach theorem, 367-370
half space, 366
Hamel basis, 89
Hausdorff spaces, 323
Heine-Borel property, 302
Heine-Borel theorem, 299
hermitian operator, 427
Hilbert space, 3 I , 377
homeomorphism, 320
homogeneous property
  of a norm, 208, 344
homogeneous system, 241-242
homomorphic image, 62,68
homomorphic rings, 67
homomorphic semigroups, 63
homomorphism, 30, 62
hyperplane, 364

I

idempotent operator, 121
identity:
  element, 35
  function, 19
  matrix, 139
  permutation, 19,44
  relation, 26
  transformation, 105,409
image of a set under f, 21
indeterminate of a
polynomial ring, 70
index:
  of a nilpotent
     operator, 185
  of a symmetric
     bilinear functional, 202
set, 10
indexed family of sets, 10
indexed set, I 1
induced:
  mapping, 20
induced
  metric, 267
  norm,349,412
  operation, 29
inequalities, 268-271
infinite-dim ensional
  vector space, 92
infinite series, 350
infinite set, 8
initial value problem, 238-261 ,328-341
injection, 14
injective, 14,100
inner product, 117,205,375
inner product space, 31,118,205
inner product subspace, 118
integral domain, 46, 49
integration:
  of matrices 249
  of vectors 249
interior, 278
intersection of sets,S
invariant linear
  subspace, 122
inverse:
  image 21
  of a function, 15, 100
  of a matrix, 140
  of an element, 38
  relation, 25
invertible element, 37
invertible linear
  transformation, 100
invertible matrix, 140
irreducible polynomial, 74
irreflexive, 372
isolated point, 275
isometric operator, 431
isometry, 321
isomorphic, 108
isomorphic semigroups, 64
isomorphism, 30, 63, 68,108

J

Jacobian matrix, 461
Jacobi identity,S 7
Jordan canonical form, 175,191

K

Kalman's theorem, 401402
kernel of a homomorphism, 65, 68
Kronecker delta, 111

L

Laplace transform, 96
latent value, 164
leading coefficient of
  a polynomial, 70
Lebesgue integral, 296
Lebesgue measurable
  function, 296
Lebesgue measurable
 sets, 295
Lebesgue measure, 295
left cancellation
  property, 34
left distributive, 28
left identity, 35
left inverse, 36
left invertible element, 37
left R-module, 54
left solution, 40
Lie algebra, 57
limit, 286
limit point, 277,288
line segment, 351
linear:
  algebra, 33
  functional, 109,355-360
  manifold,81
  operator, 31,95
  quadratic cost
     control, 468
  space, 30, 55, 76
  subspace, 59, 81, 348
  subspace generated
     by a set, 86
  transformation, 30, 95,100
  variety, 85
linearly dependent, 87
linearly independent, 87
Lipschitz condition, 324,328
lower triangular matrix, 176
lp,271
Lp ,297
L(X, Y), 104

M

map, 13
mapping, 13
mathematical system, 30
matrices, 30
matrix, 132
matrix of;
  a bilinear functional, 195
  a linear transformation, 131
  one basis with respect
     to a second basis, 149
maximal linear subspace, 363
metric, 31, 209, 264
metric space, 31,209,263-342
metric subspace, 267
minimal polynomial, 179, 181
minor of a matrix, 158
modal matrix, 172
modern algebra, 33
module, 30, 54
monic polynomial, 70
monoid,37
multiplication of a
  linear transformation
  by a scalar, 104
multiplication of
  vectors by scalars, 76,409
multiplicative semigroup, 46
multiplicity of an
  eigenvalue, 164
multivalued function, 25

N

natural basis, 126
natural coordinates, 127
n-dimensional complex
  coordinate space, 78
n-dimensional real
  coordinate space, 78
n-dimensional vector
  space, 92
negative definite
  matrix, 222
nested sequence
  of sets, 298
Neumann expansion
  theorem, 4 15
nilpotent operator, 185
non-abelian group, 40
non-commutative group, 40
non-empty set, 3
non-homogeneous system, 241-242
non-linear
  transformation, 95
non-singular linear
  transformation, 100
non-singular matriX, 140
non-void set, 3
norm, 206, 344
normal:
  equations, 395
  linear
     transformation, 237
  operator, 431
  topological space, 323
normalizing a vector, 209
normed conjugate space, 358
normed dual space, 358
normed linear space, 31, 208, 344
norm of a bounded
  linear transformation, 409
norm preserving, 367
nowhere dense, 284
null:
  matrix, 139
  set, 3
  space, 98, 224
  vector, 76, 77
nullity of a linear
  transformation, 100
n-vector, 132

O

object, 2
observations, 398
odd permutation, 156
one-to-one and onto
  mapping, 14, 100
one-to-one mapping, 14, 100
onto mapping, 14, 100
open:
  ball, 275
  covering, 299
  interval, 282
  set, 279
  sphere, 275
operation table, 27
operator, 13
optimal control problem, 468
ordered sets, 9
order of a group, 40
order of a polynomial, 70
order of a set, 8
ordinary differential
  equations, 238-261
origin, 76, 77
orthogonal:
  basis, 210
  complement, 215,382
  linear transformation, 217,231-237
  matrix, 216, 226
  projection, 123,433
  set of vectors, 379
  vectors, 118,209
orthogonality principle, 399
orthonormal set of
  vectors, 379
outcomes, 397

P

parallel, 364
parallelogram law, 208, 379
Parseval's formula, 390
Parseval's identity, 212
partial sums, 350
partitioned matrix, 147
permutation. group, 44,45
permutation on a set, 19
piecewise continuous
  derivatives, 329
point of accumulation, 277
points, 264
point spectrum, 440
polarization, 116
polynomial, 69
positive definite matrix, 222
positive operator, 429
power class, 9
power set, 9
precompact, 299
predecessor of an
  operation, 29
pre-Hilbert space, 377
primary decomposition
  theorem, 183
principal minor of a
  matrix, 158
principle of superposition, 96
probability space, 397
product metric spaces, 274
product of:
  a matrix by a scalar, 138
  linear transformations, 105,409
  two elements, 46,104
  two matrices, 138
projection, 119,226,387
projection theorem, 387,400
proper:
  subset, 3
  subspace, 81, 164
  value, 164
  vector, 164
Pythagorean theorem, 209, 379

Q

quadratic form, 115, 226
quotient, 72

R

radius, 275
random variable, 397
range of a function, 12
range of a relation, 25
range space, 98
rank of a linear
  transformation, 100
rank of a matrix, 136
rank of a symmetric
  bilinear functional, 202
real inner product space, 205
real line, 265
real vector space, 76
reduce, 435
reduced characteristic
  function, 179
reduced linear
  transformation, 122
reflection, 218
reflexive, 372
reflexive relation, 25
regular topological
  space, 323
relation, 25
relatively compact, 307
relatively prime, 73
remainder, 72
repeated eigenvalues, 173
residual, 472
residual spectrum, 440
resolution of the
  identity, 226, 457
resolvent set, 439
restriction of a mapping, 20
R-homomorphism,68
Riccati equation, 471
Riemann intergrable, 296
Riesz representation
 theorem, 393
right:
  cancellation property, 34
  distributive, 28
  identity, 34
  inverse, 35
  invertible element, 37
  R-module, 54
  solution, 40
R^∞, 78
ring, 30, 46
ring of integers, 51
ring of polynomials, 70
ring with identity, 47
R-module, 54
Rⁿ,78
rotation, 218, 230
row of a matrix, 131
row rank of a matrix, 152
row vector, 125, 132
R^*,266
R-submodule,58
R-submodule generated
  by a set, 60

S

scalar, 75
scalar multiplication, 76
Schwarz inequality, 207,376
second dual space, 371
secular value, 164
self-adjoint linear
  transformation, 221,224-225
self-adjoint operators, 428
semigroup, 30, 36
semigroup component, 46
semigroup of
  transformations, 44
semigroup operation, 46
separable, 284, 300
separates, 366
sequence, 11,286
sequence of disjoint
  sets, 12
sequence of sets, 11
sequentially compact, 301-305
set, 1
set of order zero, 8
shift operator, 441
a-algebra, 397
a-field, 397
signature of a symmetric
  bilinear functional, 202
similarity transformation, 153
similar matrices, I 53
simple eigenvalues, 164
singleton set, 8
singular linear
  transformation, 101
singular matrix, 140
skew-adjoint linear
  transformation, 221,237
skew symmetric bilinear
  functional, I 96
skew symmetric matrix, 196
skew symmetric part of a
  linear functional, 196
solution of a differential
  equation, 239
solution of an initial
  value problem, 239
space of:
  bounded complex
     sequences, 79
  bounded real sequences, 79
  finitely non-zero
     sequences, 79
  linear transformations, 104
  real-valued continuous
     functions, 80
span, 86
spectral theorem, 226,455,457
spectrum, 164,439
sphere, 275
spherical neighborhood, 275
square matrix, 132
state transition matrix, 247-255
steepest descent, 472
strictly positive, 429
strong convergence, 373
subalgebra, lOS
subcovering, 299
subdomain,52
su bfield, 52
subgroup,41
subgroup generated
  by a set, 43
submatrix, 147
subring,52
subring generated by
a set, 53
subsemigroup, 40
subsemigroup generated
  by a set, 41
subsequence, 287
subset, 3
subsystem, 40, 46
successive
  approximations, 315, 324-328
sum of:
  elements, 46
  linear operators, 409
  linear transformations, 104
  matrices, 138
  sets, 82
  vectors, 76
surjective, 14, 100
Sylvester's theorem, 199
symmetric difference
  of sets, 7
symmetric matrix, 196,226
symmetric part of a
  linear functional, 196
symmetric relation, 26
system of differential
  equations, 240, 255-260

T

ternary operation, 26
Ti-spaces,323
topological space, 31
topological structure, 31
topology, 280, 318, 322-323
totally bounded, 299
T',421
trace of a matrix, 169
transformation, 13
transformation group, 45
transitive relation, 26
transpose of a linear
  transformation, 113,420
transpose of a matrix, 133
transpose of a vector, 125
triangle inequality, 208, 264, 344
triangular matrix, 176
trivial ring, 48
trivial solution, 245
trivial subring, 53
truncation operator, 439
T^*,422
T^T,113

U

unbounded linear
  functional,356
unbounded metric space, 265
uncountable set, 23
uniform convergence, 313
uniformly continuous, 308
union of sets, 5
unit, 37
unitary operator, 431
unitary space, 205
unit of a ring, 47
unit vector, 209
unordered pair of
  elements, 9
upper triangular matrix, 176
usual metric for R*, 266, 320
usual metric on R, 265
usual metric on RTI , 271

V

vacuous set, 3
Vandermonde matrix, 260
variance, 398
vector, 75
vector addition, 75
vector space, 30, 55, 76
vector space of n-tuples
  over F, 56
vector space over a field, 76
vector subspace, 59
Venn diagram, 8
void set, 3
Volterra equation, 327
Volterra integral
  equation, 97

W

weak convergence, 373
weakly continuous, 375
weak* compact, 375
weak-star convergence, 373
Weierstrass approximation
  theorem, 285
Wronskian, 256-259

X, Y, Z

X^f ,357
X^*,357-358
zero:
  polynomial, 70
  transformation, 104,409
  vector, 76, 77
Zorn's lemma, 390