Functional analysis

X

is finitely representable in

Y

when, for every finite-dimensional subspace

E \subseteq X

and every

ε > 0

, there are a subspace

F \subseteq Y

and an isomorphism

T : E \to F

with

∥ T ∥∥ T^{- 1} ∥ < 1 + ε

Superreflexive Banach space

A Banach space is superreflexive when every Banach space finitely representable in it is a reflexive Banach space. Equivalently, it admits an equivalent uniformly convex norm, or every one of its ultrapowers is reflexive.

Convex-block separation criterion for reflexivity

X

is reflexive exactly when, for every

θ > 0

and every sequence

(x_{i})

B_{X}

, some convex combination of a finite initial segment lies within

θ

of a convex combination of the remaining tail. In a reflexive space both convex hulls approximate a common weak cluster point. The converse is the convex-block form of the James reflexivity criterion.

James reflexivity criterion

James's theorem says that a Banach space is reflexive exactly when every continuous linear functional attains its supremum on the closed unit ball. Its separation proof also yields the equivalent convex-block criterion: nonreflexivity produces a bounded sequence whose initial and tail convex hulls remain uniformly separated.

Principle of local reflexivity

For finite-dimensional subspaces

E \subseteq X^{**}

and

F \subseteq X^{*}

, the principle of local reflexivity gives an almost-isometric map

T : E \to X

that fixes

E \cap X

and preserves the pairings with

F

. It lets finite-dimensional bidual separation data be realized inside

X

Uniform finite convex-block criterion for superreflexivity

X

is superreflexive exactly when, for every

θ > 0

, some

N

forces every sequence

(x_{i})_{i = 1}^{N} \subseteq B_{X}

to have two convex combinations, one before and one after a cut, at distance below

θ

. Failure produces a nonreflexive ultrapower; finite representability transfers a violating finite sequence back from any nonreflexive space finitely representable in

X

Diamond-graph characterization of superreflexivity

X

is superreflexive exactly when the distortions

c_{X} (D_{n})

of the finite diamond graphs tend to infinity. A uniformly separated convex-block sequence recursively realizes all branches of

D_{n}

with bounded distortion, while uniform convexity forces quantitative collapse across repeated diamonds.

Bourgain embedding theorem

Every metric space with

n

points admits an embedding into a Hilbert space with distortion

O (lo g n)

. The usual random-subset construction gives an embedding into Euclidean dimension

O ((lo g n)^{2})

Frechet embedding

A Frechet embedding uses distance functions as coordinates, typically

x ⟼ (d (x, A_{i}))_{i = 1}^{k}

for selected subsets

A_{i}

. Every coordinate is one-Lipschitz, and suitable subsets provide lower bounds on pairwise image distances.

Low-dimensional Frechet embedding into linfinity ( $X ↪ ℓ_{\infty}^{k}$ )

For every integer

q \geq 2

, every

n

-point metric space embeds into

ℓ_{\infty}^{k}

with distortion at most

2 q - 1

and

k \leq Cq n^{1/ q} lo g n .

The coordinates are distance-to-subset functions. Randomly sampled subsets at

q

density scales separate every pair at one scale; a union bound leaves the stated number of coordinates.

Dimension lower bound for an expander embedded in linfinity

If a fixed-degree expander on

n

vertices embeds into

ℓ_{\infty}^{k}

with distortion

α

, then

k \geq n^{c / α} .

Indeed,

ℓ_{\infty}^{k} ↪ ℓ_{p}^{k}

has distortion

k^{1/ p}

, while

c_{p} (G) ≳ (lo g n) / p

. Taking

p ≍ lo g k

gives

lo g k ≳ (lo g n) / α

Johnson–Lindenstrauss lemma

For

0 < ε < 1

and any

n

points in a Hilbert space, there is a linear map into Euclidean dimension

O (ε^{- 2} lo g n)

that multiplies every pairwise distance by a factor between

1 - ε

and

1 + ε

Net estimate for a linear operator

The unit sphere of an

n

-dimensional normed space has a

δ

-net of size at most

(3/ δ)^{n}

. If

1 - δ \leq ∥ T x ∥ \leq 1 + δ

on this net and

δ < 1/3

, then

∥ T ∥∥ T^{- 1} ∥ \leq \frac{1 + δ}{1 - 3 δ} .

Rademacher Johnson–Lindenstrauss transform

A \in {- 1, 1}^{d \times p}

has independent Rademacher entries, then for fixed

u \neq = 0

and

0 < t < 1

P (\frac{∥ A u ∥ _{2}^{2}}{d ∥ u ∥ _{2}^{2}} - 1 \geq t) \leq 2 e^{- d t^{2} /136} .

Applying a union bound to all pairwise differences embeds

n

fixed points into dimension

O (t^{- 2} lo g (n / ε))

while preserving every squared distance within a factor

1 \pm t

with probability at least

1 - ε

Subgaussian concentration of the absolute Gaussian average

For a standard normal variable

Y

and

β = E ∣ Y ∣ = 2/ π

E e^{\pm u (∣ Y ∣ - β)} \leq e^{C u^{2}}, u \geq 0.

The exponential Markov inequality consequently gives Gaussian concentration for averages of independent copies of

∣ Y ∣

Almost-isometric Gaussian embedding from l2 into l1 ( $ℓ_{2}^{n} ↪ ℓ_{1}^{k}$ )

The random matrix

(T x)_{i} = \frac{1}{β k} \sum_{j = 1}^{n} Z_{ij} x_{j}

satisfies

P ((1 - δ) ∥ x ∥_{2} \leq ∥ T x ∥_{1} \leq (1 + δ) ∥ x ∥_{2}) \geq 1 - 2 e^{- c δ^{2} k} .

(2)

A net argument shows that

k \geq C_{ε} n

permits a linear embedding of distortion below

1 + ε

Low-dimensional L1 embedding of a finite metric space ( $X ↪ ℓ_{1}^{O (l o g n)}$ )

Every

n

-point metric space embeds into

ℓ_{1}^{O (l o g n)}

with distortion

O (lo g n)

. Apply the Bourgain embedding theorem, reduce its Euclidean dimension to

O (lo g n)

with the Johnson–Lindenstrauss lemma, and use an Almost-isometric Gaussian embedding from l2 into l1.

Exponentially weighted supremum norm ( $∥ u ∥_{α} = sup_{t} e^{- α t} ∥ u (t) ∥$ )

On continuous functions

u : [0, T] \to X

, the exponentially weighted supremum norm is

∥ u ∥_{α} = sup_{0 \leq t \leq T} e^{- α t} ∥ u (t) ∥

. It is equivalent to the ordinary supremum norm and often turns a Volterra integral map into a contraction mapping.

Operator topology

An operator topology specifies convergence of linear operators. Common choices include the norm, strong, and weak operator topologies.

Strong operator topology

Operators

T_{n}

converge to

T

in the strong operator topology when

T_{n} x \to T x

in norm for every vector

x

Weak operator topology

Operators

T_{n}

converge to

T

in the weak operator topology when

⟨ T_{n} x, y ⟩ \to ⟨ T x, y ⟩

for every pair of vectors

x, y

Operator predual from matrix coefficients

For a Hilbert space

H

, the linear span of the matrix-coefficient functionals

T \mapsto ⟨ T x, y ⟩

is a predual of

B (H)

. The unit ball is compact in the resulting weak operator topology by the Tychonoff theorem, and the compact norming dual-pair criterion identifies

B (H)

with the dual of that span.

Schauder basis

A Schauder basis of a Banach space

X

is a sequence

(e_{n})

such that every

x \in X

has a unique norm-convergent expansion

x = \sum_{n = 1}^{\infty} a_{n} e_{n}

Basis projection ( $P_{n}$ )

The nth basis projection associated with a Schauder basis is

P_{n} (\sum_{k = 1}^{\infty} a_{k} e_{k}) = \sum_{k = 1}^{n} a_{k} e_{k} .

Basis constant

The basis constant of a Schauder basis is

sup_{n} ∥ P_{n} ∥

, where

P_{n}

are its basis projections. Uniform boundedness makes this supremum finite.

Coordinate functional of a Schauder basis ( $e_{n}^{*}$ )

The nth coordinate functional of a Schauder basis is the bounded linear functional

e_{n}^{*} (\sum_{k} a_{k} e_{k}) = a_{n}

Dual sequence of a Schauder basis ( $(e_{n}^{*})$ )

The dual sequence consists of the coordinate functionals

(e_{n}^{*})

X^{*}

. It is always a basic sequence, though it need not span the entire dual space.

Shrinking Schauder basis

A Schauder basis is shrinking when its dual sequence is a Schauder basis of the whole dual space. Every Schauder basis of a reflexive Banach space is shrinking.

Basic sequence

A basic sequence in a Banach space is a sequence that is a Schauder basis for its closed linear span.

Basis selection theorem

If a bounded subset

K

of a Banach space is bounded away from zero and zero lies in its weak closure, then

K

contains a basic sequence. More generally, weak closure may be replaced by closure in any Hausdorff locally convex vector topology weaker than the weak topology.

Reflexive Banach space

X

is reflexive when its canonical embedding

J_{X} : X \to X^{**}

is surjective. Equivalently, its closed unit ball is weakly compact.

Norm minimizer in a closed convex subset of a reflexive Banach space ( $min_{x \in C} ∥ x ∥$ )

A nonempty norm-closed convex subset

C

of a reflexive Banach space attains its distance to zero. For

d = in f_{C} ∥ x ∥

, the sets

C \cap (d + 1/ n) B_{X}

are nonempty weakly closed subsets of one weakly compact ball, and form a decreasing family. Compactness gives a point in their intersection, of norm

d

. Mazur theorem supplies weak closedness of

C

and the balls; no sequential-compactness theorem is required.

Weak compactness characterization of reflexivity

A Banach space is reflexive if and only if its closed unit ball is compact in the weak topology. One direction applies the Banach-Alaoglu theorem to the unit ball in the bidual; the converse follows because Goldstine theorem makes the canonical image of the unit ball weak-star dense, while weak compactness makes that image weak-star closed.

Weak sequential compactness in a Hilbert space ( $sup_{j} ∥ u_{j} ∥_{H} < \infty ⟹ u_{j_{k}} ⇀ u$ )

Every bounded sequence in a Hilbert space has a weakly convergent subsequence. The closed span of a sequence is separable; diagonal convergence of coordinates along an orthonormal basis gives a bounded limiting linear functional, represented by a vector of the Hilbert space. Applied to a bounded family of mollifications, this proves that their strong Lebesgue limit inherits the weak derivative bound.

Weak sequential compactness of bounded sequences in a reflexive Banach space

Every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. This is the sequential form of weak compactness supplied by the Eberlein-Smulian theorem.

Weak topology

The weak topology

σ (X, X^{*})

on a normed space

X

is the coarsest topology making every member of

X^{*}

continuous. A net converges weakly exactly when every bounded linear functional converges on it.

Original and weak continuity of linear maps between Fréchet spaces ( $T : (E, τ_{E}) \to (F, τ_{F})$ )

For Fréchet spaces, the two continuity conditions for a linear map are equivalent. Original continuity makes every target continuous linear functional compose to a source continuous linear functional, proving weak continuity. Conversely, weak continuity makes these compositions continuous in the original source topology. They separate target points, so limits in the graph of a linear operator remain in the graph. The closed graph theorem for Fréchet spaces finishes the proof.

Without the hypotheses, equal continuous dual spaces do not imply equal topologies. The identity from an infinite-dimensional Hilbert space with its weak topology to its norm topology is weak-to-weak continuous but not continuous in those original topologies.

Countable separating family metrizes a weakly compact set ( $d (x, y) = \sum_{n \geq 1} 2^{- n} \frac{∣ f _{n} ( x - y ) ∣}{1 + ∣ f _{n} ( x - y ) ∣}$ )

If a countable family in

X^{*}

separates points of

X

, the displayed metric induces the weak topology on every weakly compact set

K

. The coordinate map into the countable product of scalar lines is continuous and injective; a continuous injection from a compact space into a Hausdorff space is a homeomorphism onto its image. Equivalently, uniform convergence of the metric series makes the identity from weak

K

to metric

K

continuous. Compactness is essential: point separation alone need not generate the weak topology on an arbitrary bounded set.

Weakly bounded set

A subset

D

of a normed vector space is weakly bounded when

sup_{x \in D} ∣ f (x) ∣ < \infty

for every

f \in X^{*}

. It is norm bounded: its canonical embedding into the bidual is a pointwise bounded family of functionals on the complete continuous dual space, so the Uniform boundedness principle applies. The converse is immediate. A weakly compact set is weakly bounded because each scalar evaluation has compact image.

Weak and norm Borel sigma-algebras in a separable Banach space ( $B (X, w) = B (X, ∥ \cdot ∥)$ )

The weak topology and norm topology of a separable Banach space generate the same Borel sigma-algebra. A countable norming family makes every norm ball weakly Borel measurable, and separability makes every norm-open set a countable union of such balls. The reverse inclusion follows because the weak topology is coarser.

Weak convergence ( $x_{n} ⇀ x$ )

A sequence

x_{n}

converges weakly to

x

when

f (x_{n}) \to f (x)

for every

f

in the continuous dual space

X^{*}

Radon-Riesz property

A normed vector space has the Radon-Riesz property when weak convergence

u_{k} ⇀ u

together with

∥ u_{k} ∥ \to ∥ u ∥

implies strong convergence

u_{k} \to u

. The Hilbert-space case is the Radon-Riesz theorem. The condition prevents norm from remaining in weakly invisible directions.

Finite-dimensional weak and norm topologies coincide

On a finite-dimensional vector space equipped with a norm, the weak topology and norm topology coincide. The weak topology is no finer because every element of the dual is norm-continuous. Conversely, finitely many coordinate functionals in a basis control the norm, so every sufficiently small basic weak neighbourhood lies in a prescribed norm ball.

Weakly convergent sequence is bounded

Every weakly convergent sequence in a normed vector space is a bounded sequence. Apply the Uniform boundedness principle on the Banach space

X^{*}

to the evaluation maps

J x_{n} : f \mapsto f (x_{n})

. They are pointwise bounded, while the canonical embedding into the bidual gives

∥ J x_{n} ∥ = ∥ x_{n} ∥

Metrization of the weak topology on a bounded set

Let

X^{*}

have a norm-dense sequence

(ϕ_{m})

, and let

B \subset X

be norm-bounded. Then

d (x, y) = \sum_{m = 1}^{\infty} 2^{- m} \frac{∣ ϕ _{m} ( x - y ) ∣}{1 + ∣ ϕ _{m} ( x - y ) ∣}

is a metric on

B

that induces its weak topology. Density lets convergence against the

ϕ_{m}

extend uniformly on

B - B

to convergence against every

ϕ \in X^{*}

Weak closure

The weak closure

\overline{K}^{w}

K

is its closure in the weak topology. Thus

x \in \overline{K}^{w}

exactly when every weak neighbourhood of

x

meets

K

Weakly null sequence

A sequence

(x_{n})

is weakly null when

f (x_{n}) \to 0

for every member

f

of the continuous dual space. It may remain bounded away from zero in norm.

Standard unit vectors are weakly null in lp ( $e_{n} ⇀ 0$ in $ℓ^{p}$ )

For

1 < p < \infty

, identify

(ℓ^{p})^{*}

with

ℓ^{q}

. If

y = (y_{j}) \in ℓ^{q}

, then

⟨ y, e_{n} ⟩ = y_{n} ⟶ 0.

Thus the standard unit vectors converge weakly to zero in

ℓ^{p}

, although

∥ e_{n} ∥_{p} = 1

for every

n

Weak convergence of bounded disjointly supported sequences in lp ( $ℓ^{p}$ )

Let

1 < p < \infty

. Every norm-bounded sequence in

ℓ^{p}

whose terms have pairwise disjoint supports converges weakly to zero. Indeed, the average of

N

distinct terms has norm

O (N^{1/ p - 1})

, so Mazur theorem gives weak convergence through its convex-combination criterion.

Weakly Cauchy sequence

A sequence

(x_{n})

is weakly Cauchy when

(f (x_{n}))

is a Cauchy sequence for every continuous linear functional

f

. Its consecutive differences form a weakly null sequence.

Weak closure of the unit sphere

In an infinite-dimensional normed space, the weak closure of the unit sphere is the closed unit ball. Every basic weak neighbourhood imposes only finitely many linear conditions, whose common kernel contains a nonzero direction that can move any interior point onto the sphere.

Weak-star topology

The weak-star topology

σ (X^{*}, X)

is pointwise convergence of functionals on

X

. If

X

is nonreflexive, it is strictly weaker than the weak topology

σ (X^{*}, X^{**})

X^{*}

Weak-star separability of the entire dual

The continuous dual of a separable Banach space is separable in its weak-star topology, even when it is not norm separable. Banach-Alaoglu theorem and weak-star metrizability of the dual ball make the dual unit ball a separable compact metric space. The countable union of positive integer multiples of a countable dense subset of that ball is dense in the whole dual. This does not imply global weak-star metrizability.

Weak-star open dual ball category obstruction ( ${ℓ \in E^{*} : ∥ ℓ ∥ < 1} = ⋃_{n} {∥ ℓ ∥ \leq 1 - 1/ (n + 1)}$ )

For an infinite-dimensional normed space, the weak-star open unit ball of its dual is not a Baire space. The smaller closed norm balls are weak-star closed and have empty relative interior: a nonzero functional annihilating any finitely many tested vectors can move the norm beyond the smaller radius while remaining below one. Their countable union is the open norm ball. Thus this topology is not topologically complete, regardless of separability.

Weak-star topology on an entire infinite-dimensional Banach dual is not metrizable

A countable local base at zero would give countably many finite sets of evaluation vectors. Any other evaluation must lie in the span of one of those finite sets: otherwise a bounded functional vanishing on that set but not on the evaluation vector, supplied by Hahn-Banach theorem, contradicts inclusion of neighbourhoods after scalar rescaling. Thus the predual has countable Hamel dimension, impossible for an infinite-dimensional Banach space by the Baire category theorem. This concerns the entire dual, unlike weak-star metrizability of the dual ball.

Continuous dual of a weak-star topology ( $(X^{}, σ (X^{}, X))^{'} = J_{X} (X)$ )

A linear functional on

X^{*}

is continuous for the weak-star topology exactly when it is evaluation at an element of

X

. Continuity makes it vanish on the common kernel of finitely many evaluations, so it is a linear combination of those evaluations. The canonical embedding into the bidual identifies this dual with

X

Failure of weak-star convergence to commute with squaring

L^{\infty} (R) = (L^{1} (R))^{*}

, the sequence

f_{n} (x) = sin (n x)

converges weak-star to zero by the Riemann-Lebesgue lemma, while

f_{n}^{2}

converges weak-star to the constant

1/2

. Nonlinear pointwise operations therefore need not preserve weak-star limits.

Weak-star metrizability of the dual ball

X

is separable, the weak-star topology on

B_{X^{*}}

is metrizable. For a dense sequence

(x_{n})

B_{X}

, a compatible metric is obtained by summing bounded multiples of

2^{- n} ∣ f (x_{n}) - g (x_{n}) ∣

Weak-star metrizability criterion for a dual ball

The dual unit ball

B_{X^{*}}

is weak-star metrizable if and only if

X

is separable. For the converse direction, Banach-Alaoglu theorem makes a metrizable dual ball a compact metric space, and the evaluation map embeds

X

isometrically into the separable space

C (B_{X^{*}})

Grothendieck space

X

is a Grothendieck space when every weak-star convergent sequence in

X^{*}

is weakly convergent. Every reflexive Banach space has this property, and every separable Grothendieck space is reflexive.

Szlenk derivation ( $K_{ε}^{'}$ )

For a weak-star compact set

K \subseteq X^{*}

, its Szlenk derivation at scale

ε > 0

K_{ε}^{'} = {f \in K : diam (U \cap K) > ε for every weak-star neighbourhood U of f} .

It removes points having a relatively weak-star open neighbourhood of norm diameter at most

ε

One-step Szlenk derivation for a separable dual

X^{*}

is separable and

K

is a nonempty weak-star closed subset of

B_{X^{*}}

, then

K_{ε}^{'}

is a proper subset of

K

for every

ε > 0

. Metrizability produces sequences witnessing membership in the derivative, while a Baire-category argument applied to a universal weakly null sequence yields the strict inclusion.

Weakly compact set

A subset of a Banach space is weakly compact when it is compact in the weak topology; it is relatively weakly compact when its weak closure is weakly compact.

Discontinuous pointwise limit obstruction to weak compactness

A bounded sequence in the space of continuous functions on a compact space that has a discontinuous pointwise limit cannot lie in a weakly compact set. Compactness would give a weakly convergent subnet, and continuous point-evaluation functionals would identify its continuous limit with the prescribed pointwise limit. No assumption of sequential compactness is needed.

Weakly compact subsets of l-infinity are norm separable

Coordinate evaluations on the l-infinity sequence space form a countable separating family, so a weakly compact set

K

is metrizable by countable separating family metrizes a weakly compact set. Choose a countable weakly dense subset

D

. Then

K

lies in the weak closure of

conv D

, which equals its norm closure by Mazur theorem. Rational convex combinations make that norm closure separable, and every subset of a separable metric space is separable. The weakly dense subset itself need not be norm dense in a nonconvex

K

Weakly compact set is norm bounded

Every weakly compact set in a normed vector space is norm bounded, even if the space is incomplete. Its canonical embedding into the bidual is pointwise bounded on the continuous dual space; the Uniform boundedness principle on that Banach dual gives the norm bound.

∥ ab ∥ \leq ∥ a ∥ ∥ b ∥

Semisimple commutative Banach algebra ( $rad A = 0$ )

A commutative Banach algebra is semisimple when its Jacobson radical is zero. For a complex unital Banach algebra this is equivalent to injectivity of the Gelfand transform. This condition concerns the Jacobson radical; it does not assert a finite product decomposition as in finite-dimensional semisimple algebra theory.

Supremum bound for a complete function-algebra norm ( $∥ f ∥_{\infty} \leq ∥ f ∥$ )

For an algebra of complex-valued functions with a complete algebra norm, every point value

f (t)

belongs to the spectrum of an element of

f

in the artificial unitization. Indeed

(h, λ) \mapsto h (t) + λ

is an algebraic unital character of an algebra, and it vanishes on

f - f (t) 1

. The spectrum bound gives

∣ f (t) ∣ \leq ∥ f ∥

without first assuming that evaluation is continuous.

Group of invertible elements of a Banach algebra ( $G (A)$ )

The invertible elements of a unital Banach algebra form an open multiplicative group. For

a \in G (A)

, every

b

satisfying

∥ a^{- 1} ∥∥ b - a ∥ < 1

is invertible, because

b = a (1 + a^{- 1} (b - a))

and the second factor has a Neumann series inverse.

Continuity of inversion in a Banach algebra ( $a \mapsto a^{- 1}$ )

Inversion is continuous on the group of invertible elements of a Banach algebra. The Neumann series gives a local bound on inverse norms, while

b^{- 1} - a^{- 1} = b^{- 1} (a - b) a^{- 1}

makes the difference tend to zero as

b \to a

Qp-Banach algebra ( $Q_{p}$ -Banach algebra)

A Qp-Banach algebra is a

Q_{p}

-algebra with a complete non-Archimedean submultiplicative norm satisfying

∥ λa ∥ = ∣ λ ∣_{p} ∥ a ∥

Gelfand-Mazur theorem

Every complex unital Banach algebra in which every nonzero element is invertible is isometrically isomorphic to

C

Algebra norm

An algebra norm is a norm satisfying

∥ ab ∥ \leq ∥ a ∥ ∥ b ∥

. Its completion is a Banach algebra in which the original algebra embeds densely.

Minimality of the supremum norm on C(K) ( $∥ f ∥_{\infty} \leq ∥ f ∥_{1}$ )

Any submultiplicative algebra norm on all of

C (K)

dominates its supremum norm, even when the new norm is incomplete. Every evaluation character extends continuously to the completion: the restrictions of the completion's character space form a closed subset of

K

, and an omitted point would give a nonzero function annihilated by an invertible function in the completion.

Unitization of an algebra

The unitization adjoins an identity to a possibly nonunital algebra. For

C_{0} (X)

it is naturally

C (X^{+})

on the one-point compactification.

Neumann series

a

belongs to a unital Banach algebra and

∥ a ∥ < 1

, then

(1 - a)^{- 1} = \sum_{n = 0}^{\infty} a^{n} .

Absolute convergence and multiplication of partial sums prove the identity.

Spectrum of an element ( $σ_{A} (a)$ )

The spectrum of

a

in a unital complex algebra

A

σ_{A} (a) = {λ \in C : a - λ 1 is not invertible} .

For a nonunital algebra it is defined in the unitization of an algebra.

Nonemptiness of the Banach-algebra spectrum

Every element of a nonzero complex unital Banach algebra has a nonempty compact spectrum of an element. If the resolvent of an element existed everywhere, composing it with any bounded linear functional would give a bounded entire function tending to zero at infinity. The Liouville theorem and point separation by the Hahn-Banach theorem would force the resolvent to be zero, contradicting the inverse equation. Nonunital algebras are treated by unitization.

Approximate point spectrum ( $σ_{ap} (T)$ )

The approximate point spectrum consists of the scalars

λ

for which there are unit vectors

x_{k}

satisfying

∥(T - λ I) x_{k} ∥ \to 0

. Equivalently,

T - λ I

is not bounded below.

Bilateral shift operator ( $T : ℓ^{2} (Z) \to ℓ^{2} (Z)$ )

The bilateral shift

(T x)_{n} = x_{n + 1}

is unitary. It has empty point spectrum, while its approximate point spectrum is the unit circle. For

∣ λ ∣ = 1

, normalized vectors with entries

λ^{n}

on an interval of length

N

fail to satisfy

T x = λ x

only at the two endpoints, giving residual norm

2/ N

Resolvent of an element ( $R (λ, a)$ )

The resolvent is

R (λ, a) = (λ 1 - a)^{- 1}

on the complement of the spectrum of an element. In a Banach algebra it is analytic there.

Resolvent identity

For two points

z, w

in the resolvent set of an operator

A

R (z, A) - R (w, A) = (w - z) R (z, A) R (w, A) .

In particular, the resolvent operator is differentiable and

\partial_{z} R (z, A) = - R (z, A)^{2}

Resolvent norm ( $∥(A - z I)^{- 1} ∥$ )