Let V denote a Banach space over the field F. A Schauder basis is a sequence {b_{n}} of elements of V such that for every element v ∈ V there exists a unique sequence {α_{n}} of scalars in F so that
v
=
∑
n
=
0
∞
α
n
b
n
,
where the convergence is understood with respect to the norm topology, i.e.,
lim
n
→
∞
∥
v
−
∑
k
=
0
n
α
k
b
k
∥
V
=
0.
Schauder bases can also be defined analogously in a general topological vector space. As opposed to a Hamel basis, the elements of the basis must be ordered since the series may not converge unconditionally.
A Schauder basis {b_{n}}_{ n ≥ 0} is said to be normalized when all the basis vectors have norm 1 in the Banach space V.
A sequence {x_{n}}_{ n ≥ 0} in V is a basic sequence if it is a Schauder basis of its closed linear span.
Two Schauder bases, {b_{n}} in V and {c_{n}} in W, are said to be equivalent if there exist two constants c > 0 and C such that for every integer N ≥ 0 and all sequences {α_{n}} of scalars,
c
∥
∑
k
=
0
N
α
k
b
k
∥
V
≤
∥
∑
k
=
0
N
α
k
c
k
∥
W
≤
C
∥
∑
k
=
0
N
α
k
b
k
∥
V
.
A family of vectors in V is total if its linear span (the set of finite linear combinations) is dense in V. If V is a Hilbert space, an orthogonal basis is a total subset B of V such that elements in B are nonzero and pairwise orthogonal. Further, when each element in B has norm 1, then B is an orthonormal basis of V.
Let {b_{n}} be a Schauder basis of a Banach space V over F = R or C. It follows from the Banach–Steinhaus theorem that the linear mappings {P_{n}} defined by
v
=
∑
k
=
0
∞
α
k
b
k
⟶
P
n
P
n
(
v
)
=
∑
k
=
0
n
α
k
b
k
are uniformly bounded by some constant C. When C = 1, the basis is called a monotone basis. The maps {P_{n}} are the basis projections.
Let {b*_{n}} denote the coordinate functionals, where b*_{n} assigns to every vector v in V the coordinate α_{n} of v in the above expansion. Each b*_{n} is a bounded linear functional on V. Indeed, for every vector v in V,

b
n
∗
(
v
)

∥
b
n
∥
V
=

α
n

∥
b
n
∥
V
=
∥
α
n
b
n
∥
V
=
∥
P
n
(
v
)
−
P
n
−
1
(
v
)
∥
V
≤
2
C
∥
v
∥
V
.
These functionals {b*_{n}} are called biorthogonal functionals associated to the basis {b_{n}}. When the basis {b_{n}} is normalized, the coordinate functionals {b*_{n}} have norm ≤ 2C in the continuous dual V ′ of V.
A Banach space with a Schauder basis is necessarily separable, but the converse is false, as described below. Since every vector v in a Banach space V with a Schauder basis is the limit of P_{n}(v), with P_{n} of finite rank and uniformly bounded, such a space V satisfies the bounded approximation property.
A theorem attributed to Mazur asserts that every infinitedimensional Banach space V contains a basic sequence, i.e., there is an infinitedimensional subspace of V that has a Schauder basis. The basis problem is the question asked by Banach, whether every separable Banach space has a Schauder basis. This was negatively answered by Per Enflo who constructed a separable Banach space failing the approximation property, thus a space without a Schauder basis.
The standard unit vector bases of c_{0}, and of ℓ^{p} for 1 ≤ p < ∞, are monotone Schauder bases. In this unit vector basis {b_{n}}, the vector b_{n} in V = c_{0} or in V = ℓ^{p} is the scalar sequence {b_{ n, j }}_{ j} where all coordinates b_{n, j} are 0, except the nth coordinate:
b
n
=
{
b
n
,
j
}
j
=
0
∞
∈
V
,
b
n
,
j
=
δ
n
,
j
,
where δ_{n, j} is the Kronecker delta. The space ℓ^{∞} is not separable, and therefore has no Schauder basis.
Every orthonormal basis in a separable Hilbert space is a Schauder basis. Every countable orthonormal basis is equivalent to the standard unit vector basis in ℓ^{2}.
The Haar system is an example of a basis for L^{p}([0, 1]), when 1 ≤ p < ∞. When 1 < p < ∞, another example is the trigonometric system defined below. The Banach space C([0, 1]) of continuous functions on the interval [0, 1], with the supremum norm, admits a Schauder basis. The Faber–Schauder system is the most commonly used Schauder basis for C([0, 1]).
Several bases for classical spaces were discovered before Banach's book appeared (Banach (1932)), but some other cases remained open for a long time. For example, the question of whether the disk algebra A(D) has a Schauder basis remained open for more than forty years, until Bočkarev showed in 1974 that a basis constructed from the Franklin system exists in A(D). One can also prove that the periodic Franklin system is a basis for a Banach space A_{r} isomorphic to A(D). This space A_{r} consists of all complex continuous functions on the unit circle T whose conjugate function is also continuous. The Franklin system is another Schauder basis for C([0, 1]), and it is a Schauder basis in L^{p}([0, 1]) when 1 ≤ p < ∞. Systems derived from the Franklin system give bases in the space C^{1}([0, 1]^{2}) of differentiable functions on the unit square. The existence of a Schauder basis in C^{1}([0, 1]^{2}) was a question from Banach's book.
Let {x_{n}} be, in the real case, the sequence of functions
{
1
,
cos
(
x
)
,
sin
(
x
)
,
cos
(
2
x
)
,
sin
(
2
x
)
,
cos
(
3
x
)
,
sin
(
3
x
)
,
…
}
or, in the complex case,
{
1
,
e
i
x
,
e
−
i
x
,
e
2
i
x
,
e
−
2
i
x
,
e
3
i
x
,
e
−
3
i
x
,
…
}
.
The sequence {x_{n}} is called the trigonometric system. It is a Schauder basis for the space L^{p}([0, 2π]) for any p such that 1 < p < ∞. For p = 2, this is the content of the Riesz–Fischer theorem, and for p ≠ 2, it is a consequence of the boundedness on the space L^{p}([0, 2π]) of the Hilbert transform on the circle. It follows from this boundedness that the projections P_{N} defined by
{
f
:
x
→
∑
k
=
−
∞
+
∞
c
k
e
i
k
x
}
⟶
P
N
{
P
N
f
:
x
→
∑
k
=
−
N
N
c
k
e
i
k
x
}
are uniformly bounded on L^{p}([0, 2π]) when 1 < p < ∞. This family of maps {P_{N}} is equicontinuous and tends to the identity on the dense subset consisting of trigonometric polynomials. It follows that P_{N} f tends to f in L^{p}norm for every f ∈ L^{p}([0, 2π]). In other words, {x_{n}} is a Schauder basis of L^{p}([0, 2π]).
However, the set {x_{n}} is not a Schauder basis for L^{1}([0, 2π]). This means that there are functions in L^{1} whose Fourier series does not converge in the L^{1} norm, or equivalently, that the projections P_{N} are not uniformly bounded in L^{1}norm. Also, the set {x_{n}} is not a Schauder basis for C([0, 2π]).
The space K(ℓ^{2}) of compact operators on the Hilbert space ℓ^{2} has a Schauder basis. For every x, y in ℓ^{2}, let x ⊗ y denote the rank one operator v ∈ ℓ^{2} → <v, x> y. If {e_{n }}_{n ≥ 1} is the standard orthonormal basis of ℓ^{2}, a basis for K(ℓ^{2}) is given by the sequence
e
1
⊗
e
1
,
e
1
⊗
e
2
,
e
2
⊗
e
2
,
e
2
⊗
e
1
,
…
,
e
1
⊗
e
n
,
e
2
⊗
e
n
,
…
,
e
n
⊗
e
n
,
e
n
⊗
e
n
−
1
,
…
,
e
n
⊗
e
1
,
…
For every n, the sequence consisting of the n^{2} first vectors in this basis is a suitable ordering of the family {e_{j} ⊗ e_{k}}, for 1 ≤ j, k ≤ n.
The preceding result can be generalized: a Banach space X with a basis has the approximation property, so the space K(X) of compact operators on X is isometrically isomorphic to the injective tensor product
X
′
⊗
^
ε
X
≃
K
(
X
)
.
If X is a Banach space with a Schauder basis {e_{n }}_{n ≥ 1} such that the biorthogonal functionals are a basis of the dual, that is to say, a Banach space with a shrinking basis, then the space K(X) admits a basis formed by the rank one operators e *_{j} ⊗ e_{k} : v → e *_{j }(v) e_{k}, with the same ordering as before. This applies in particular to every reflexive Banach space X with a Schauder basis
On the other hand, the space B(ℓ^{2}) has no basis, since it is nonseparable. Moreover, B(ℓ^{2}) does not have the approximation property.
A Schauder basis {b_{n}} is unconditional if whenever the series
∑
α
n
b
n
converges, it converges unconditionally. For a Schauder basis {b_{n}}, this is equivalent to the existence of a constant C such that
∥
∑
k
=
0
n
ε
k
α
k
b
k
∥
V
≤
C
∥
∑
k
=
0
n
α
k
b
k
∥
V
for all integers n, all scalar coefficients {α_{k}} and all signs ε_{k} = ± 1. Unconditionality is an important property since it allows one to forget about the order of summation. A Schauder basis is symmetric if it is unconditional and uniformly equivalent to all its permutations: there exists a constant C such that for every integer n, every permutation π of the integers {0, 1, …, n} , all scalar coefficients {α_{k}} and all signs {ε_{k}},
∥
∑
k
=
0
n
ε
k
α
k
b
π
(
k
)
∥
V
≤
C
∥
∑
k
=
0
n
α
k
b
k
∥
V
.
The standard bases of the sequence spaces c_{0} and ℓ^{p} for 1 ≤ p < ∞, as well as every orthonormal basis in a Hilbert space, are unconditional. These bases are also symmetric.
The trigonometric system is not an unconditional basis in L^{p}, except for p = 2.
The Haar system is an unconditional basis in L^{p} for any 1 < p < ∞. The space L^{1}([0, 1]) has no unconditional basis.
A natural question is whether every infinitedimensional Banach space has an infinitedimensional subspace with an unconditional basis. This was solved negatively by Timothy Gowers and Bernard Maurey in 1992.
A basis {e_{n}}_{n≥0} of a Banach space X is boundedly complete if for every sequence {a_{n}}_{n≥0} of scalars such that the partial sums
V
n
=
∑
k
=
0
n
a
k
e
k
are bounded in X, the sequence {V_{n}} converges in X. The unit vector basis for ℓ^{p}, 1 ≤ p < ∞, is boundedly complete. However, the unit vector basis is not boundedly complete in c_{0}. Indeed, if a_{n} = 1 for every n, then
∥
V
n
∥
c
0
=
max
0
≤
k
≤
n

a
k

=
1
for every n, but the sequence {V_{n}} is not convergent in c_{0}, since V_{n+1} − V_{n} = 1 for every n.
A space X with a boundedly complete basis {e_{n}}_{n≥0} is isomorphic to a dual space, namely, the space X is isomorphic to the dual of the closed linear span in the dual X ′ of the biorthogonal functionals associated to the basis {e_{n}}.
A basis {e_{n}}_{n≥0} of X is shrinking if for every bounded linear functional f on X, the sequence of nonnegative numbers
φ
n
=
sup
{

f
(
x
)

:
x
∈
F
n
,
∥
x
∥
≤
1
}
tends to 0 when n → ∞, where F_{n} is the linear span of the basis vectors e_{m} for m ≥ n. The unit vector basis for ℓ^{p}, 1 < p < ∞, or for c_{0}, is shrinking. It is not shrinking in ℓ^{1}: if f is the bounded linear functional on ℓ^{1} given by
f
:
x
=
{
x
n
}
∈
ℓ
1
→
∑
n
=
0
∞
x
n
,
then φ_{n} ≥ f(e_{n}) = 1 for every n.
A basis {e_{n }}_{n ≥ 0} of X is shrinking if and only if the biorthogonal functionals {e*_{n }}_{n ≥ 0} form a basis of the dual X ′.
Robert C. James characterized reflexivity in Banach spaces with basis: the space X with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete. James also proved that a space with an unconditional basis is nonreflexive if and only if it contains a subspace isomorphic to c_{0} or ℓ^{1}.
A Hamel basis is a subset B of a vector space V such that every element v ∈ V can uniquely be written as
v
=
∑
b
∈
B
α
b
b
with α_{b} ∈ F, with the extra condition that the set
{
b
∈
B
∣
α
b
≠
0
}
is finite. This property makes the Hamel basis unwieldy for infinitedimensional Banach spaces; as a Hamel basis for an infinitedimensional Banach space has to be uncountable. (Every finitedimensional subspace of an infinitedimensional Banach space X has empty interior, and is nowhere dense in X. It then follows from the Baire category theorem that a countable union of these finitedimensional subspaces cannot serve as a basis.)