(Rd), 233B(x, r), 18
C(X, Y), 67
CL(X), 67
CL(X, Y), 67
C[a, b], 3
Cn[a, b], 17
Dk, 231
H(x), 234
Hn(a, b), 242
K(X, Y), 211
L(X), 57
L(X, Y), 57
(Rd), 233
Lp(Ω), 14
O(2), 50
[·, ·], 84
D′(Rd), 233
D(Rd), 230
L1(A), 368
S′(R), 255
S(R), 254
ℓ∞, 4
ℓp, 4
S2, 21
Sd−1, 50
{·, ·}, 152
c, 5
c0, 5
p-adic norm, 17
absolute time, 141
absolutely convergent series, 37
absolutely integrable function, 367
action, 146
algebra, 83
almost all, 363
almost everywhere, 363
Appollonius identity, 162
approximate spectrum, 103
arc length, 61
Archimedean property, 21
averaging operator, 77
Axiom of Choice, 113
Baire Lemma, 93
Banach algebra, 83
Banach limit, 116
Banach space, 27
Banach, Stefan, 27
Banach-Steinhauss Theorem, 97
Bernstein polynomials, 22
Bernstein, Sergei, 22
Besicovitch almost periodic function, 188
Bessel’s inequality, 172
bidual of a normed space, 110
Bolzano-Weierstrass Theorem, 30
bounded above, 112
bounded linear operator, 73
bounded variation, 105
Cantor set, 47
Cauchy sequence, 26
Cauchy-Schwarz inequality, 157
Cayley transform, 199
Cesáro summation, 77
chain, 112
chain rule, 125
classical mechanics, 145
closed ball, 20
Closed Graph Theorem, 96
closed interval, 20
closed unit ball, 16
closure (of a set), 43
coarser topology, 79
commutant, 252
complementary observables, 205
complete normed space, 27
completion of an inner product space, 163
composition of maps, 64
composition of operators, 82
conservation of angular momentum, 143
continuity at a point, 60
continuous map, 60
continuously differentiable function, 4
convergence in D, 232
convergence in S, 254
convergent sequence, 24
convex function, 128
convolution, 257
Convolution Theorem, 256
d’Alembert’s formula, 56
delta distribution, 235
delta function, 229
derivative, 119
diagonalizable matrix, 91
differentiability at a point, 119
differentiable, 119
diffusion equation, 90
dipole, 238
Dirac delta distribution, 235
Dirac distribution, 103
Dirichlet problem, 86
discrete topology, 79
distance, 8
distribution, 233
distributional derivative, 237
dual operator, 106
dual space of a normed space, 104
eigenvalue, 98
eigenvector, 98
energy, 148
Enflo, Per, 77
epigraph, 129
equivalence class, 163
equivalence relation, 163
equivalent norms, 33
essential supremum norm, 14
Euclidean norm, 9
Euclidean plane, 141
Euler-Lagrange equation, 134
Extreme Value Theorem, 10
factorial of a multi-index, 249
finite rank operator, 211
Fréchet derivative, 119
Fredholm integral equation, 87
functional analysis, vii
functional, bounded linear, 104
fundamental solution, 241, 242, 251, 252, 257
Fundamental Theorem of ODEs, 39
Galerkin approximation, 219
Galerkin method, 217
Galilean spacetime, 141
Gateux derivative, 122
Gelfand-Beurling formula, 102
generalised Fourier series, 185
Gram-Schmidt orthonormalisation, 167
graph, 96
Green’s function, 257
Hahn-Banach Lemma, 111
Hahn-Banach Theorem, 108
Hamiltonian, 150
Hamiltonian equations, 151
Hamiltonian mechanics, 150
harmonic oscillator, 172
Heaviside function, 234
Heaviside, Oliver, 234
Heisenberg’s uncertainty principle, 84, 204
helicoid, 144
Hermite functions, 172
Hermitian operator, 195
Hilbert cube, 48
Hilbert space, 160
Hilbert-Schmidt norm, 76, 163, 196
Hilbert-Schmidt operator, 196, 215
hyperbolic rotation, 50
hypoelliptic differential operator, 252
ideal of an algebra, 216
identity operator, 83
image of a set, 62
index, 20
index set, 20
induced norm, by an inner product, 157
induced norm, in a subspace, 15
inner product, 156
inner product space, 156
integrable function, 367
integral equation, 87
integral operator, 70
invariant subspace, 77
invariant subspace problem, 77
inverse image of a set, 62
invertible operator, 84
isometric isomorphism, 185
isomorphic normed spaces, 96
isomorphism, 96
Isoperimetric Theorem, 187
Jacobi identity, 153
Jordan canonical form, 92
jump rule, 239
Kepler’s second law, 143
kernel of an integral operator, 71
kinetic energy, 146
Lagrangian, 146
Lagrangian density, 143
Laplace equation, 171, 241, 242, 257
Laplacian, 242
law of conservation of energy, 148
law of conservation of momentum, 148
least squares approximation problem, 174
least squares regression, 176
left shift operator, 54, 68, 102
Leibniz rule, 249
Leibniz’s formula, 169
Lie algebra, 152
limit point of a set, 43
linear isometric embedding, 110
linear transformation, 54
Lipschitz condition, 39
Lipschitz function, 42
locally integrable function, 233
matrix mechanics, 200
maximal element, 113
measurable function, 363
measurable set, 360
measure, 360
metric space, 8
mid-point convexity, 94
minimising sequence, 178
minimum surface area, 144
Minkowski spacetime, 141
module, 250
momentum, 148
multi-index notation, 231
multiplication, of a distribution by a smooth function, 249
multiplicative identity element, 83
Neumann series, 86
Neumann Series Theorem, 86
Neumann, Carl, 86
Newton’s second law, 145
Newtonian mechanics, 145
nilpotent matrix, 92
Noether’s Theorem, 148
norm, 8
normed algebra, 83
normed space, 8
numerical analysis, 217
observables, 84
ODE, 39
open ball, 17
open cover, 51
open interval, 18
Open Mapping Theorem, 95
open operator, 93
open set, 18
open set in a topological space, 23
open unit ball, 16
operator norm, 74
order of a multi-index, 249
ordinary differential equation, 39
orthogonal complement, 172
orthogonal projection of a vector, 180
orthonormal basis, 183
parallelogram law, 160
Parseval’s identity, 186
partial order on a set, 112
partially ordered set, 112
partition of an interval, 105
perturbation theory, 237
phase plane, 151
phase plane trajectory, 151
phase portrait, 151
Poisson bracket, 152
Poissonian mechanics, 152
polarisation formula, 161
positive operator, 226
potential energy, 145
principle of stationary action, 145
product of normed spaces, 44
projection approximation, 220
proper time, 142
Pythagoras’s Theorem, 161
quantum mechanics, 84, 103, 153, 171, 172, 200, 229, 237, 251
reflexive space, 110
regular distribution, 234
relation, 163
representation theorems, 104
resolvent, 99
resolvent identity, 102
Riesz Representation Theorem, 104, 189
Riesz, Marcel, 189
Rodrigues’s formula, 169
scalar multiplication, 3
Schauder basis, 76
Schrödinger equation, 172
Schwartz space of test functions, 254
seaview property, 28
self-adjoint operator, 195
separable normed space, 21
separable space, 77
simple function, 365
skew-adjoint operator, 195
Sloan approximation, 220
smaller topology, 79
Sobolev spaces, 242
special relativity, 142
Spectral Mapping Theorem, 103
spectral radius, 102
Spectral Theorem for Compact Operators, 103
spectrum, 99
sphere, 50
square root operator, 226
state, 151
stationary for a functional, curve/function/solution, 134
step function, 364
strong operator topology, 80
stronger topology, 79
subcover, 51
subspace, 4
supremum norm, 10
symmetric set, 94
tempered distribution, 254
test function, 230
topological space, 23
topology, 23
total variation, 105
trivial topology, 79
twin paradox, 142
unbounded operator, 103
Uniform Boundedness Principle, 96
uniform operator topology, 80
unit sphere, 21
unitary operator, 195
upper bound, 112
vector addition, 3
vector space, 3
wave mechanics, 200
weak operator topology, 81
weaker topology, 79
Weierstrass’s Approximation Theorem, 22
Weierstrass’s Theorem, 66
zero linear transformation, 57
zero vector, 3