In mathematics, more specifically in functional analysis, a Banach space is a
In mathematics, more specifically in functional analysis, a Banach space is a

complete normed vector space. Thus, a Banach space is a vector space with a
complete normed vector space. Thus, a Banach space is a vector space with a

metric that allows the computation of vector length and distance between
metric that allows the computation of vector length and distance between

vectors and is complete in the sense that a Cauchy sequence of vectors always
vectors and is complete in the sense that a Cauchy sequence of vectors always

converges to a well defined limit that is within the space.
converges to a well defined limit that is within the space.

Banach spaces are named after the Polish mathematician Stefan Banach, who
Banach spaces are named after the Polish mathematician Stefan Banach, who

introduced and made a systematic study of them in 1920–1922 along with Hans
introduced and made a systematic study of them in 1920–1922 along with Hans

Hahn and Eduard Helly. Banach spaces originally grew out of the study of
Hahn and Eduard Helly. Banach spaces originally grew out of the study of

function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach
function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach

spaces play a central role in functional analysis. In other areas of analysis,
spaces play a central role in functional analysis. In other areas of analysis,

the spaces under study are often Banach spaces.
the spaces under study are often Banach spaces.

Definition A Banach space is a vector space X over
Definition A Banach space is a vector space X over

the field R of real numbers, or over the field C of complex numbers, which is
the field R of real numbers, or over the field C of complex numbers, which is

equipped with a norm and which is complete with respect to that norm, that
equipped with a norm and which is complete with respect to that norm, that

is to say, for every Cauchy sequence {xn} in X, there exists an element x in
is to say, for every Cauchy sequence {xn} in X, there exists an element x in

X such that or equivalently:
X such that or equivalently:

The vector space structure allows one to relate the behavior of Cauchy sequences
The vector space structure allows one to relate the behavior of Cauchy sequences

to that of converging series of vectors. A normed space X is a Banach space if
to that of converging series of vectors. A normed space X is a Banach space if

and only if each absolutely convergent series in X converges,
and only if each absolutely convergent series in X converges,

Completeness of a normed space is preserved if the given norm is replaced
Completeness of a normed space is preserved if the given norm is replaced

by an equivalent one. All norms on a finite-dimensional vector
by an equivalent one. All norms on a finite-dimensional vector

space are equivalent. Every finite-dimensional normed space over R
space are equivalent. Every finite-dimensional normed space over R

or C is a Banach space. General theory
or C is a Banach space. General theory

= Linear operators, isomorphisms= If X and Y are normed spaces over the
= Linear operators, isomorphisms= If X and Y are normed spaces over the

same ground field K, the set of all continuous K-linear maps T : X → Y is
same ground field K, the set of all continuous K-linear maps T : X → Y is

denoted by B(X, Y). In infinite-dimensional spaces, not all
denoted by B(X, Y). In infinite-dimensional spaces, not all

linear maps are continuous. A linear mapping from a normed space X to another
linear maps are continuous. A linear mapping from a normed space X to another

normed space is continuous if and only if it is bounded on the closed unit ball
normed space is continuous if and only if it is bounded on the closed unit ball

of X. Thus, the vector space B(X, Y) can be given the operator norm
of X. Thus, the vector space B(X, Y) can be given the operator norm

For Y a Banach space, the space B(X, Y) is a Banach space with respect to this
For Y a Banach space, the space B(X, Y) is a Banach space with respect to this

norm. If X is a Banach space, the space B(X) =
norm. If X is a Banach space, the space B(X) =

B(X, X) forms a unital Banach algebra; the multiplication operation is given by
B(X, X) forms a unital Banach algebra; the multiplication operation is given by

the composition of linear maps. If X and Y are normed spaces, they are
the composition of linear maps. If X and Y are normed spaces, they are

isomorphic normed spaces if there exists a linear bijection T : X → Y such that T
isomorphic normed spaces if there exists a linear bijection T : X → Y such that T

and its inverse T −1 are continuous. If one of the two spaces X or Y is complete
and its inverse T −1 are continuous. If one of the two spaces X or Y is complete

then so is the other space. Two normed spaces X and Y are isometrically
then so is the other space. Two normed spaces X and Y are isometrically

isomorphic if in addition, T is an isometry, i.e., ||T(x)|| = ||x|| for
isomorphic if in addition, T is an isometry, i.e., ||T(x)|| = ||x|| for

every x in X. The Banach-Mazur distance d(X, Y) between two isomorphic but not
every x in X. The Banach-Mazur distance d(X, Y) between two isomorphic but not

isometric spaces X and Y gives a measure of how much the two spaces X and Y
isometric spaces X and Y gives a measure of how much the two spaces X and Y

differ. = Basic notions=
differ. = Basic notions=

Every normed space X can be isometrically embedded in a Banach
Every normed space X can be isometrically embedded in a Banach

space. More precisely, there is a Banach space Y and an isometric mapping T : X →
space. More precisely, there is a Banach space Y and an isometric mapping T : X →

Y such that T(X) is dense in Y. If Z is another Banach space such that there is
Y such that T(X) is dense in Y. If Z is another Banach space such that there is

an isometric isomorphism from X onto a dense subset of Z, then Z is
an isometric isomorphism from X onto a dense subset of Z, then Z is

isometrically isomorphic to Y. This Banach space Y is the completion of
isometrically isomorphic to Y. This Banach space Y is the completion of

the normed space X. The underlying metric space for Y is the same as the
the normed space X. The underlying metric space for Y is the same as the

metric completion of X, with the vector space operations extended from X to Y.
metric completion of X, with the vector space operations extended from X to Y.

The completion of X is often denoted by The cartesian product X × Y of two
The completion of X is often denoted by The cartesian product X × Y of two

normed spaces is not canonically equipped with a norm. However, several
normed spaces is not canonically equipped with a norm. However, several

equivalent norms are commonly used, such as
equivalent norms are commonly used, such as

and give rise to isomorphic normed spaces. In this sense, the product X × Y
and give rise to isomorphic normed spaces. In this sense, the product X × Y

is complete if and only if the two factors are complete.
is complete if and only if the two factors are complete.

If M is a closed linear subspace of a normed space X, there is a natural norm
If M is a closed linear subspace of a normed space X, there is a natural norm

on the quotient space X / M, The quotient X / M is a Banach space
on the quotient space X / M, The quotient X / M is a Banach space

when X is complete. The quotient map from X onto X / M, sending x in X to its
when X is complete. The quotient map from X onto X / M, sending x in X to its

class x + M, is linear, onto and has norm 1, except when M = X, in which case
class x + M, is linear, onto and has norm 1, except when M = X, in which case

the quotient is the null space. The closed linear subspace M of X is
the quotient is the null space. The closed linear subspace M of X is

said to be a complemented subspace of X if M is the range of a bounded linear
said to be a complemented subspace of X if M is the range of a bounded linear

projection P from X onto M. In this case, the space X is isomorphic to the
projection P from X onto M. In this case, the space X is isomorphic to the

direct sum of M and Ker(P), the kernel of the projection P.
direct sum of M and Ker(P), the kernel of the projection P.

Suppose that X and Y are Banach spaces and that T ∈ B(X, Y). There exists a
Suppose that X and Y are Banach spaces and that T ∈ B(X, Y). There exists a

canonical factorization of T as where the first map π is the quotient
canonical factorization of T as where the first map π is the quotient

map, and the second map T1 sends every class x + Ker(T) in the quotient to the
map, and the second map T1 sends every class x + Ker(T) in the quotient to the

image T(x) in Y. This is well defined because all elements in the same class
image T(x) in Y. This is well defined because all elements in the same class

have the same image. The mapping T1 is a linear bijection from X / Ker(T) onto
have the same image. The mapping T1 is a linear bijection from X / Ker(T) onto

the range T(X), whose inverse need not be bounded.
the range T(X), whose inverse need not be bounded.

= Classical spaces= Basic examples of Banach spaces include:
= Classical spaces= Basic examples of Banach spaces include:

the Lp spaces and their special cases, the sequence spaces ℓp that consist of
the Lp spaces and their special cases, the sequence spaces ℓp that consist of

scalar sequences indexed by N; among them, the space ℓ1 of absolutely
scalar sequences indexed by N; among them, the space ℓ1 of absolutely

summable sequences and the space ℓ2 of square summable sequences; the space c0
summable sequences and the space ℓ2 of square summable sequences; the space c0

of sequences tending to zero and the space ℓ∞ of bounded sequences; the space
of sequences tending to zero and the space ℓ∞ of bounded sequences; the space

C(K) of continuous scalar functions on a compact Hausdorff space K, equipped with
C(K) of continuous scalar functions on a compact Hausdorff space K, equipped with

the max norm, According to the Banach–Mazur theorem,
the max norm, According to the Banach–Mazur theorem,

every Banach space is isometrically isomorphic to a subspace of some C(K).
every Banach space is isometrically isomorphic to a subspace of some C(K).

For every separable Banach space X, there is a closed subspace M of ℓ1 such
For every separable Banach space X, there is a closed subspace M of ℓ1 such

that X ≅ ℓ1/M. Any Hilbert space serves as an example
that X ≅ ℓ1/M. Any Hilbert space serves as an example

of a Banach space. A Hilbert space H on K = R, C is complete for a norm of the
of a Banach space. A Hilbert space H on K = R, C is complete for a norm of the

form where
form where

is the inner product, linear in its first argument that satisfies the
is the inner product, linear in its first argument that satisfies the

following: For example, the space L2 is a Hilbert
following: For example, the space L2 is a Hilbert

space. The Hardy spaces, the Sobolev spaces are
space. The Hardy spaces, the Sobolev spaces are

examples of Banach spaces that are related to Lp spaces and have additional
examples of Banach spaces that are related to Lp spaces and have additional

structure. They are important in different branches of analysis, Harmonic
structure. They are important in different branches of analysis, Harmonic

analysis and Partial differential equations among others.
analysis and Partial differential equations among others.

= Banach algebras= A Banach algebra is a Banach space A
= Banach algebras= A Banach algebra is a Banach space A

over K = R or C, together with a structure of algebra over K, such that
over K = R or C, together with a structure of algebra over K, such that

the product map ∈ A × A → ab ∈ A is continuous. An equivalent norm on A can
the product map ∈ A × A → ab ∈ A is continuous. An equivalent norm on A can

be found so that ||ab|| ≤ ||a|| ||b|| for all a, b ∈ A.
be found so that ||ab|| ≤ ||a|| ||b|| for all a, b ∈ A.

Examples The Banach space C(K), with the
Examples The Banach space C(K), with the

pointwise product, is a Banach algebra. The disk algebra A(D) consists of
pointwise product, is a Banach algebra. The disk algebra A(D) consists of

functions holomorphic in the open unit disk D ⊂ C and continuous on its
functions holomorphic in the open unit disk D ⊂ C and continuous on its

closure: D. Equipped with the max norm on D, the disk algebra A(D) is a closed
closure: D. Equipped with the max norm on D, the disk algebra A(D) is a closed

subalgebra of C(D). The Wiener algebra A(T) is the algebra
subalgebra of C(D). The Wiener algebra A(T) is the algebra

of functions on the unit circle T with absolutely convergent Fourier series.
of functions on the unit circle T with absolutely convergent Fourier series.

Via the map associating a function on T to the sequence of its Fourier
Via the map associating a function on T to the sequence of its Fourier

coefficients, this algebra is isomorphic to the Banach algebra ℓ1(Z), where the
coefficients, this algebra is isomorphic to the Banach algebra ℓ1(Z), where the

product is the convolution of sequences. For every Banach space X, the space B(X)
product is the convolution of sequences. For every Banach space X, the space B(X)

of bounded linear operators on X, with the composition of maps as product, is a
of bounded linear operators on X, with the composition of maps as product, is a

Banach algebra. A C*-algebra is a complex Banach algebra
Banach algebra. A C*-algebra is a complex Banach algebra

A with an antilinear involution a → a∗ such that ||a∗a|| = ||a||2. The space
A with an antilinear involution a → a∗ such that ||a∗a|| = ||a||2. The space

B(H) of bounded linear operators on a Hilbert space H is a fundamental example
B(H) of bounded linear operators on a Hilbert space H is a fundamental example

of C*-algebra. The Gelfand–Naimark theorem states that every C*-algebra is
of C*-algebra. The Gelfand–Naimark theorem states that every C*-algebra is

isometrically isomorphic to a C*-subalgebra of some B(H). The space
isometrically isomorphic to a C*-subalgebra of some B(H). The space

C(K) of complex continuous functions on a compact Hausdorff space K is an
C(K) of complex continuous functions on a compact Hausdorff space K is an

example of commutative C*-algebra, where the involution associates to every
example of commutative C*-algebra, where the involution associates to every

function  f  its complex conjugate  f . = Dual space=
function  f  its complex conjugate  f . = Dual space=

If X is a normed space and K the underlying field, the continuous dual
If X is a normed space and K the underlying field, the continuous dual

space is the space of continuous linear maps from X into K, or continuous linear
space is the space of continuous linear maps from X into K, or continuous linear

functionals. The notation for the continuous dual is X ′ = B(X, K) in this
functionals. The notation for the continuous dual is X ′ = B(X, K) in this

article. Since K is a Banach space, the dual X ′ is a Banach space, for every
article. Since K is a Banach space, the dual X ′ is a Banach space, for every

normed space X. The main tool for proving the existence
normed space X. The main tool for proving the existence

of continuous linear functionals is the Hahn–Banach theorem.
of continuous linear functionals is the Hahn–Banach theorem.

Hahn–Banach theorem. Let X be a vector space over the field K = R, C. Let
Hahn–Banach theorem. Let X be a vector space over the field K = R, C. Let

further Y ⊆ X be a linear subspace,
further Y ⊆ X be a linear subspace,

p : X → R be a sublinear function and  f  : Y → K be a linear functional so
p : X → R be a sublinear function and  f  : Y → K be a linear functional so

that Re( f (y)) ≤ p(y) for all y in Y. Then, there exists a linear functional F
that Re( f (y)) ≤ p(y) for all y in Y. Then, there exists a linear functional F

: X → K so that In particular, every continuous linear
: X → K so that In particular, every continuous linear

functional on a subspace of a normed space can be continuously extended to
functional on a subspace of a normed space can be continuously extended to

the whole space, without increasing the norm of the functional. An important
the whole space, without increasing the norm of the functional. An important

special case is the following: for every vector x in a normed space X, there
special case is the following: for every vector x in a normed space X, there

exists a continuous linear functional  f  on X such that
exists a continuous linear functional  f  on X such that

When x is not equal to the 0 vector, the functional  f  must have norm one, and
When x is not equal to the 0 vector, the functional  f  must have norm one, and

is called a norming functional for x. The Hahn–Banach separation theorem
is called a norming functional for x. The Hahn–Banach separation theorem

states that two disjoint non-empty convex sets in a real Banach space, one
states that two disjoint non-empty convex sets in a real Banach space, one

of them open, can be separated by a closed affine hyperplane. The open
of them open, can be separated by a closed affine hyperplane. The open

convex set lies strictly on one side of the hyperplane, the second convex set
convex set lies strictly on one side of the hyperplane, the second convex set

lies on the other side but may touch the hyperplane.
lies on the other side but may touch the hyperplane.

A subset S in a Banach space X is total if the linear span of S is dense in X.
A subset S in a Banach space X is total if the linear span of S is dense in X.

The subset S is total in X if and only if the only continuous linear functional
The subset S is total in X if and only if the only continuous linear functional

that vanishes on S is the 0 functional: this equivalence follows from the
that vanishes on S is the 0 functional: this equivalence follows from the

Hahn–Banach theorem. If X is the direct sum of two closed
Hahn–Banach theorem. If X is the direct sum of two closed

linear subspaces M and N, then the dual X ′ of X is isomorphic to the direct sum
linear subspaces M and N, then the dual X ′ of X is isomorphic to the direct sum

of the duals of M and N. If M is a closed linear subspace in X, one can
of the duals of M and N. If M is a closed linear subspace in X, one can

associate the orthogonal of M in the dual,
associate the orthogonal of M in the dual,

The orthogonal M ⊥ is a closed linear subspace of the dual. The dual of M is
The orthogonal M ⊥ is a closed linear subspace of the dual. The dual of M is

isometrically isomorphic to X ′ / M ⊥. The dual of X / M is isometrically
isometrically isomorphic to X ′ / M ⊥. The dual of X / M is isometrically

isomorphic to M ⊥. The dual of a separable Banach space
isomorphic to M ⊥. The dual of a separable Banach space

need not be separable, but: Theorem. Let X be a normed space. If X ′
need not be separable, but: Theorem. Let X be a normed space. If X ′

is separable, then X is separable. When X ′ is separable, the above
is separable, then X is separable. When X ′ is separable, the above

criterion for totality can be used for proving the existence of a countable
criterion for totality can be used for proving the existence of a countable

total subset in X. Weak topologies
total subset in X. Weak topologies

The weak topology on a Banach space X is the coarsest topology on X for which all
The weak topology on a Banach space X is the coarsest topology on X for which all

elements x ′ in the continuous dual space X ′ are continuous. The norm
elements x ′ in the continuous dual space X ′ are continuous. The norm

topology is therefore finer than the weak topology. It follows from the
topology is therefore finer than the weak topology. It follows from the

Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a
Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a

norm-closed convex subset of a Banach space is also weakly closed. A
norm-closed convex subset of a Banach space is also weakly closed. A

norm-continuous linear map between two Banach spaces X and Y is also weakly
norm-continuous linear map between two Banach spaces X and Y is also weakly

continuous, i.e., continuous from the weak topology of X to that of Y.
continuous, i.e., continuous from the weak topology of X to that of Y.

If X is infinite-dimensional, there exist linear maps which are not
If X is infinite-dimensional, there exist linear maps which are not

continuous. The space X∗ of all linear maps from X to the underlying field K
continuous. The space X∗ of all linear maps from X to the underlying field K

also induces a topology on X which is finer than the weak topology, and much
also induces a topology on X which is finer than the weak topology, and much

less used in functional analysis. On a dual space X ′, there is a topology
less used in functional analysis. On a dual space X ′, there is a topology

weaker than the weak topology of X ′, called weak* topology. It is the
weaker than the weak topology of X ′, called weak* topology. It is the

coarsest topology on X ′ for which all evaluation maps x′ ∈ X ′ → x′(x), x ∈ X,
coarsest topology on X ′ for which all evaluation maps x′ ∈ X ′ → x′(x), x ∈ X,

are continuous. Its importance comes from the Banach–Alaoglu theorem.
are continuous. Its importance comes from the Banach–Alaoglu theorem.

Banach–Alaoglu Theorem. Let X be a normed vector space. Then the closed
Banach–Alaoglu Theorem. Let X be a normed vector space. Then the closed

unit ball B ′ = {x′ ∈ X ′ : ||x′|| ≤ 1} of the dual space is compact in the
unit ball B ′ = {x′ ∈ X ′ : ||x′|| ≤ 1} of the dual space is compact in the

weak* topology. The Banach–Alaoglu theorem depends on
weak* topology. The Banach–Alaoglu theorem depends on

Tychonoff's theorem about infinite products of compact spaces. When X is
Tychonoff's theorem about infinite products of compact spaces. When X is

separable, the unit ball B ′ of the dual is a metrizable compact in the weak*
separable, the unit ball B ′ of the dual is a metrizable compact in the weak*

topology. Examples of dual spaces
topology. Examples of dual spaces

The dual of c0 is isometrically isomorphic to ℓ1: for every bounded
The dual of c0 is isometrically isomorphic to ℓ1: for every bounded

linear functional  f  on c0, there is a unique element y = {yn} ∈ ℓ1 such that
linear functional  f  on c0, there is a unique element y = {yn} ∈ ℓ1 such that

The dual of ℓ1 is isometrically isomorphic to ℓ∞. The dual of Lp([0, 1])
The dual of ℓ1 is isometrically isomorphic to ℓ∞. The dual of Lp([0, 1])

is isometrically isomorphic to Lq([0, 1]) when 1 ≤ p is symmetric, and in the
is isometrically isomorphic to Lq([0, 1]) when 1 ≤ p is symmetric, and in the

complex case, that it satisfies the Hermitian symmetry property and = i .
complex case, that it satisfies the Hermitian symmetry property and = i .

The parallelogram law implies that is additive in x. It follows that it is
The parallelogram law implies that is additive in x. It follows that it is

linear over the rationals, thus linear by continuity.
linear over the rationals, thus linear by continuity.

Several characterizations of spaces isomorphic to Hilbert spaces are
Several characterizations of spaces isomorphic to Hilbert spaces are

available. The parallelogram law can be extended to more than two vectors, and
available. The parallelogram law can be extended to more than two vectors, and

weakened by the introduction of a two-sided inequality with a constant c ≥
weakened by the introduction of a two-sided inequality with a constant c ≥

1: Kwapień proved that if for every integer n and all families of
1: Kwapień proved that if for every integer n and all families of

vectors {x1, ..., xn} ⊂ X, then the Banach space X is isomorphic to a
vectors {x1, ..., xn} ⊂ X, then the Banach space X is isomorphic to a

Hilbert space. Here, Ave± denotes the average over the 2n possible choices of
Hilbert space. Here, Ave± denotes the average over the 2n possible choices of

signs ±1. In the same article, Kwapień proved that the validity of a
signs ±1. In the same article, Kwapień proved that the validity of a

Banach-valued Parseval's theorem for the Fourier transform characterizes Banach
Banach-valued Parseval's theorem for the Fourier transform characterizes Banach

spaces isomorphic to Hilbert spaces. Lindenstrauss and Tzafriri proved that a
spaces isomorphic to Hilbert spaces. Lindenstrauss and Tzafriri proved that a

Banach space in which every closed linear subspace is complemented is
Banach space in which every closed linear subspace is complemented is

isomorphic to a Hilbert space. The proof rests upon Dvoretzky's theorem about
isomorphic to a Hilbert space. The proof rests upon Dvoretzky's theorem about

Euclidean sections of high-dimensional centrally symmetric convex bodies. In
Euclidean sections of high-dimensional centrally symmetric convex bodies. In

other words, Dvoretzky's theorem states that for every integer n, any
other words, Dvoretzky's theorem states that for every integer n, any

finite-dimensional normed space, with dimension sufficiently large compared to
finite-dimensional normed space, with dimension sufficiently large compared to

n, contains subspaces nearly isometric to the n-dimensional Euclidean space.
n, contains subspaces nearly isometric to the n-dimensional Euclidean space.

The next result gives the solution of the so-called homogeneous space problem.
The next result gives the solution of the so-called homogeneous space problem.

An infinite-dimensional Banach space X is said to be homogeneous if it is
An infinite-dimensional Banach space X is said to be homogeneous if it is

isomorphic to all its infinite-dimensional closed subspaces. A
isomorphic to all its infinite-dimensional closed subspaces. A

Banach space isomorphic to ℓ2 is homogeneous, and Banach asked for the
Banach space isomorphic to ℓ2 is homogeneous, and Banach asked for the

converse. Theorem. A Banach space isomorphic to
converse. Theorem. A Banach space isomorphic to

all its infinite-dimensional closed subspaces is isomorphic to a separable
all its infinite-dimensional closed subspaces is isomorphic to a separable

Hilbert space. An infinite-dimensional Banach space is
Hilbert space. An infinite-dimensional Banach space is

hereditarily indecomposable when no subspace of it can be isomorphic to the
hereditarily indecomposable when no subspace of it can be isomorphic to the

direct sum of two infinite-dimensional Banach spaces. The Gowers dichotomy
direct sum of two infinite-dimensional Banach spaces. The Gowers dichotomy

theorem asserts that every infinite-dimensional Banach space X
theorem asserts that every infinite-dimensional Banach space X

contains, either a subspace Y with unconditional basis, or a hereditarily
contains, either a subspace Y with unconditional basis, or a hereditarily

indecomposable subspace Z, and in particular, Z is not isomorphic to its
indecomposable subspace Z, and in particular, Z is not isomorphic to its

closed hyperplanes. If X is homogeneous, it must therefore have an unconditional
closed hyperplanes. If X is homogeneous, it must therefore have an unconditional

basis. It follows then from the partial solution obtained by Komorowski and
basis. It follows then from the partial solution obtained by Komorowski and

Tomczak–Jaegermann, for spaces with an unconditional basis, that X is
Tomczak–Jaegermann, for spaces with an unconditional basis, that X is

isomorphic to ℓ2. = Spaces of continuous functions=
isomorphic to ℓ2. = Spaces of continuous functions=

When two compact Hausdorff spaces K1 and K2 are homeomorphic, the Banach spaces
When two compact Hausdorff spaces K1 and K2 are homeomorphic, the Banach spaces

C(K1) and C(K2) are isometric. Conversely, when K1 is not homeomorphic
C(K1) and C(K2) are isometric. Conversely, when K1 is not homeomorphic

to K2, the Banach–Mazur distance between C(K1) and C(K2) must be greater than or
to K2, the Banach–Mazur distance between C(K1) and C(K2) must be greater than or

equal to 2, see above the results by Amir and Cambern. Although uncountable
equal to 2, see above the results by Amir and Cambern. Although uncountable

compact metric spaces can have different homeomorphy types, one has the following
compact metric spaces can have different homeomorphy types, one has the following

result due to Milutin: Theorem. Let K be an uncountable compact
result due to Milutin: Theorem. Let K be an uncountable compact

metric space. Then C(K) is isomorphic to C([0, 1]).
metric space. Then C(K) is isomorphic to C([0, 1]).

The situation is different for countably infinite compact Hausdorff spaces. Every
The situation is different for countably infinite compact Hausdorff spaces. Every

countably infinite compact K is homeomorphic to some closed interval of
countably infinite compact K is homeomorphic to some closed interval of

ordinal numbers equipped with the order topology, where
ordinal numbers equipped with the order topology, where

α is a countably infinite ordinal. The Banach space C(K) is then isometric to
α is a countably infinite ordinal. The Banach space C(K) is then isometric to

C(). When α, β are two countably infinite ordinals, and assuming α ≤ β,
C(). When α, β are two countably infinite ordinals, and assuming α ≤ β,

the spaces C() and C() are isomorphic if and only if β
the spaces C() and C() are isomorphic if and only if β