diff --git a/src/sequence_spaces/arithmetic/action.jl b/src/sequence_spaces/arithmetic/action.jl
index 1aaccb9a..d4dffcab 100644
--- a/src/sequence_spaces/arithmetic/action.jl
+++ b/src/sequence_spaces/arithmetic/action.jl
@@ -1,8 +1,8 @@
 # fallback methods
 
-(A::LinearOperator)(b::Sequence) = *(A, b)
+(A::LinearOperator)(b::AbstractSequence) = *(A, b)
 
-function Base.:*(A::LinearOperator, b::Sequence)
+function Base.:*(A::LinearOperator, b::AbstractSequence)
     domain_A, codomain_A = domain(A), codomain(A)
     space_b = space(b)
     _iscompatible(domain_A, space_b) || return throw(ArgumentError("spaces must be compatible: A has domain $domain_A, b has space $space_b"))
@@ -12,13 +12,13 @@ function Base.:*(A::LinearOperator, b::Sequence)
     return c
 end
 
-function mul!(c::Sequence, A::LinearOperator, b::Sequence, α::Number, β::Number)
+function mul!(c::Sequence, A::LinearOperator, b::AbstractSequence, α::Number, β::Number)
     _iscompatible(space(c), codomain(A)) & _iscompatible(domain(A), space(b)) || return throw(ArgumentError("spaces must be compatible"))
     _mul!(c, A, b, α, β)
     return c
 end
 
-function _mul!(c::Sequence, A::LinearOperator, b::Sequence, α::Number, β::Number)
+function _mul!(c::Sequence, A::LinearOperator, b::AbstractSequence, α::Number, β::Number)
     domain_A, codomain_A = domain(A), codomain(A)
     space_b = space(b)
     space_c = space(c)
@@ -51,7 +51,7 @@ function _mul!(c::Sequence, A::LinearOperator, b::Sequence, α::Number, β::Numb
     return c
 end
 
-function Base.:\(A::LinearOperator, b::Sequence)
+function Base.:\(A::LinearOperator, b::AbstractSequence)
     codomain_A = codomain(A)
     space_b = space(b)
     _iscompatible(codomain_A, space_b) || return throw(ArgumentError("spaces must be compatible: A has codomain $codomain_A, b has space $space_b"))
diff --git a/src/sequence_spaces/arithmetic/fft.jl b/src/sequence_spaces/arithmetic/fft.jl
index 6c75c836..a15749b4 100644
--- a/src/sequence_spaces/arithmetic/fft.jl
+++ b/src/sequence_spaces/arithmetic/fft.jl
@@ -10,9 +10,9 @@ fft_size(s₁::TensorSpace{<:NTuple{N,BaseSpace}}, s₂::TensorSpace{<:NTuple{N,
 fft_size(s₁::TensorSpace{<:NTuple{N,BaseSpace}}, s₂::TensorSpace{<:NTuple{N,BaseSpace}}, s₃::TensorSpace{<:NTuple{N,BaseSpace}}...) where {N} =
     ntuple(i -> nextpow(2, _dfs_dimension(s₁, i) + _dfs_dimension(s₂, i) + mapreduce(s₃_ -> _dfs_dimension(s₃_, i), +, s₃)), Val(N))
 
-fft_size(s::BaseSpace, n::Int) = (_fft_length(s, n),)
+fft_size(s::BaseSpace, n::Integer) = (_fft_length(s, n),)
 
-fft_size(s::TensorSpace, n::Int) = map(sᵢ -> _fft_length(sᵢ, n), spaces(s))
+fft_size(s::TensorSpace, n::Integer) = map(sᵢ -> _fft_length(sᵢ, n), spaces(s))
 
 _fft_length(s₁::BaseSpace, s₂::BaseSpace) =
     nextpow(2, _dfs_dimension(s₁) + _dfs_dimension(s₂))
@@ -20,13 +20,13 @@ _fft_length(s₁::BaseSpace, s₂::BaseSpace) =
 _fft_length(s₁::BaseSpace, s₂::BaseSpace, s₃::BaseSpace...) =
     nextpow(2, _dfs_dimension(s₁) + _dfs_dimension(s₂) + mapreduce(_dfs_dimension, +, s₃))
 
-_fft_length(s::BaseSpace, n::Int) = nextpow(2, n * _dfs_dimension(s))
+_fft_length(s::BaseSpace, n::Integer) = nextpow(2, n * _dfs_dimension(s))
 
 #
 
 _dfs_dimensions(s::TensorSpace) = map(_dfs_dimension, spaces(s))
 
-_dfs_dimension(s::TensorSpace, i::Int) = _dfs_dimension(s[i])
+_dfs_dimension(s::TensorSpace, i::Integer) = _dfs_dimension(s[i])
 
 # Taylor
 _dfs_dimension(s::Taylor) = dimension(s)
@@ -37,9 +37,9 @@ _dfs_dimension(s::Chebyshev) = 2order(s)+1
 
 # sequence to discrete Fourier series
 
-fft(a::Sequence{<:BaseSpace}, n::Tuple{Int}) = @inbounds fft(a, n[1])
+fft(a::Sequence{<:BaseSpace}, n::Tuple{Integer}) = @inbounds fft(a, n[1])
 
-function fft(a::Sequence{<:BaseSpace}, n::Int)
+function fft(a::Sequence{<:BaseSpace}, n::Integer)
     space_a = space(a)
     _is_fft_size_compatible(n, space_a) || return throw(DimensionMismatch)
     CoefType = complex(eltype(a))
@@ -62,7 +62,7 @@ function fft!(C::AbstractVector, a::Sequence{<:BaseSpace})
     return _fft_pow2!(C)
 end
 
-function fft(a::Sequence{TensorSpace{T}}, n::NTuple{N,Int}) where {N,T<:NTuple{N,BaseSpace}}
+function fft(a::Sequence{TensorSpace{T}}, n::NTuple{N,Integer}) where {N,T<:NTuple{N,BaseSpace}}
     space_a = space(a)
     _is_fft_size_compatible(n, space_a) || return throw(DimensionMismatch)
     CoefType = complex(eltype(a))
diff --git a/src/sequence_spaces/arithmetic/linear_operator.jl b/src/sequence_spaces/arithmetic/linear_operator.jl
index 64ff9128..57894ebd 100644
--- a/src/sequence_spaces/arithmetic/linear_operator.jl
+++ b/src/sequence_spaces/arithmetic/linear_operator.jl
@@ -80,7 +80,7 @@ function lmul!(A::LinearOperator, B::LinearOperator)
     return LinearOperator(domain(B), codomain(A), lmul!(coefficients(A), coefficients(B)))
 end
 
-function Base.:^(A::LinearOperator, n::Int)
+function Base.:^(A::LinearOperator, n::Integer)
     if n < 0
         return ^(inv(A), -n)
     elseif n == 0
diff --git a/src/sequence_spaces/special_operators/calculus.jl b/src/sequence_spaces/special_operators/calculus.jl
index 1273f3e5..f379f64f 100644
--- a/src/sequence_spaces/special_operators/calculus.jl
+++ b/src/sequence_spaces/special_operators/calculus.jl
@@ -89,24 +89,24 @@ Integral(order::Int...) = Integral(order)
 order(ℐ::Integral) = ℐ.order
 
 """
-    *(𝒟::Derivative, a::Sequence)
+    *(𝒟::Derivative, a::AbstractSequence)
 
 Compute the `order(𝒟)`-th derivative of `a`; equivalent to `differentiate(a, order(𝒟))`.
 
-See also: [`(::Derivative)(::Sequence)`](@ref), [`Derivative`](@ref),
+See also: [`(::Derivative)(::AbstractSequence)`](@ref), [`Derivative`](@ref),
 [`differentiate`](@ref) and [`differentiate!`](@ref).
 """
-Base.:*(𝒟::Derivative, a::Sequence) = differentiate(a, order(𝒟))
+Base.:*(𝒟::Derivative, a::AbstractSequence) = differentiate(a, order(𝒟))
 
 """
-    (𝒟::Derivative)(a::Sequence)
+    (𝒟::Derivative)(a::AbstractSequence)
 
 Compute the `order(𝒟)`-th derivative of `a`; equivalent to `differentiate(a, order(𝒟))`.
 
-See also: [`*(::Derivative, ::Sequence)`](@ref), [`Derivative`](@ref),
+See also: [`*(::Derivative, ::AbstractSequence)`](@ref), [`Derivative`](@ref),
 [`differentiate`](@ref) and [`differentiate!`](@ref).
 """
-(𝒟::Derivative)(a::Sequence) = *(𝒟, a)
+(𝒟::Derivative)(a::AbstractSequence) = *(𝒟, a)
 
 """
     differentiate(a::Sequence, α=1)
@@ -179,24 +179,24 @@ function project!(C::LinearOperator, 𝒟::Derivative)
 end
 
 """
-    *(ℐ::Integral, a::Sequence)
+    *(ℐ::Integral, a::AbstractSequence)
 
 Compute the `order(ℐ)`-th integral of `a`; equivalent to `integrate(a, order(ℐ))`.
 
-See also: [`(::Integral)(::Sequence)`](@ref), [`Integral`](@ref),
+See also: [`(::Integral)(::AbstractSequence)`](@ref), [`Integral`](@ref),
 [`integrate`](@ref) and [`integrate!`](@ref).
 """
-Base.:*(ℐ::Integral, a::Sequence) = integrate(a, order(ℐ))
+Base.:*(ℐ::Integral, a::AbstractSequence) = integrate(a, order(ℐ))
 
 """
-    (ℐ::Integral)(a::Sequence)
+    (ℐ::Integral)(a::AbstractSequence)
 
 Compute the `order(ℐ)`-th integral of `a`; equivalent to `integrate(a, order(ℐ))`.
 
-See also: [`*(::Integral, ::Sequence)`](@ref), [`Integral`](@ref),
+See also: [`*(::Integral, ::AbstractSequence)`](@ref), [`Integral`](@ref),
 [`integrate`](@ref) and [`integrate!`](@ref).
 """
-(ℐ::Integral)(a::Sequence) = *(ℐ, a)
+(ℐ::Integral)(a::AbstractSequence) = *(ℐ, a)
 
 """
     integrate(a::Sequence, α=1)
@@ -273,9 +273,9 @@ for (F, f) ∈ ((:Derivative, :differentiate), (:Integral, :integrate))
         Base.:*(ℱ₁::$F{Int}, ℱ₂::$F{Int}) = $F(order(ℱ₁) + order(ℱ₂))
         Base.:*(ℱ₁::$F{NTuple{N,Int}}, ℱ₂::$F{NTuple{N,Int}}) where {N} = $F(map(+, order(ℱ₁), order(ℱ₂)))
 
-        Base.:^(ℱ::$F{Int}, n::Int) = $F(order(ℱ) * n)
-        Base.:^(ℱ::$F{<:Tuple{Vararg{Int}}}, n::Int) = $F(map(αᵢ -> *(αᵢ, n), order(ℱ)))
-        Base.:^(ℱ::$F{NTuple{N,Int}}, n::NTuple{N,Int}) where {N} = $F(map(*, order(ℱ), n))
+        Base.:^(ℱ::$F{Int}, n::Integer) = $F(order(ℱ) * n)
+        Base.:^(ℱ::$F{<:Tuple{Vararg{Int}}}, n::Integer) = $F(map(αᵢ -> *(αᵢ, n), order(ℱ)))
+        Base.:^(ℱ::$F{NTuple{N,Int}}, n::NTuple{N,Integer}) where {N} = $F(map(*, order(ℱ), n))
 
         _findposition_nzind_domain(ℱ::$F, domain, codomain) =
             _findposition(_nzind_domain(ℱ, domain, codomain), domain)
diff --git a/src/sequence_spaces/special_operators/evaluation.jl b/src/sequence_spaces/special_operators/evaluation.jl
index b55c4069..139d8ddb 100644
--- a/src/sequence_spaces/special_operators/evaluation.jl
+++ b/src/sequence_spaces/special_operators/evaluation.jl
@@ -38,36 +38,36 @@ Evaluation(value::Union{Number,Nothing}...) = Evaluation(value)
 value(ℰ::Evaluation) = ℰ.value
 
 """
-    *(ℰ::Evaluation, a::Sequence)
+    *(ℰ::Evaluation, a::AbstractSequence)
 
 Evaluate `a` at `value(ℰ)`; equivalent to `evaluate(a, value(ℰ))`.
 
-See also: [`(::Evaluation)(::Sequence)`](@ref), [`Evaluation`](@ref),
-[`(::Sequence)(::Any, ::Vararg)`](@ref), [`evaluate`](@ref) and [`evaluate!`](@ref).
+See also: [`(::Evaluation)(::AbstractSequence)`](@ref), [`Evaluation`](@ref),
+[`(::AbstractSequence)(::Any, ::Vararg)`](@ref), [`evaluate`](@ref) and [`evaluate!`](@ref).
 """
-Base.:*(ℰ::Evaluation, a::Sequence) = evaluate(a, value(ℰ))
+Base.:*(ℰ::Evaluation, a::AbstractSequence) = evaluate(a, value(ℰ))
 
 """
-    (ℰ::Evaluation)(a::Sequence)
+    (ℰ::Evaluation)(a::AbstractSequence)
 
 Evaluate `a` at `value(ℰ)`; equivalent to `evaluate(a, value(ℰ))`.
 
-See also: [`*(::Evaluation, ::Sequence)`](@ref), [`Evaluation`](@ref),
-[`(::Sequence)(::Any, ::Vararg)`](@ref), [`evaluate`](@ref) and [`evaluate!`](@ref).
+See also: [`*(::Evaluation, ::AbstractSequence)`](@ref), [`Evaluation`](@ref),
+[`(::AbstractSequence)(::Any, ::Vararg)`](@ref), [`evaluate`](@ref) and [`evaluate!`](@ref).
 """
-(ℰ::Evaluation)(a::Sequence) = *(ℰ, a)
+(ℰ::Evaluation)(a::AbstractSequence) = *(ℰ, a)
 
 """
-    (a::Sequence)(x, y...)
+    (a::AbstractSequence)(x, y...)
 
 Evaluate `a` at `(x, y...)`; equivalent to `evaluate(a, (x, y...))` or
 `evaluate(a, x)` if `y` is not provided.
 
 See also: [`evaluate`](@ref), [`evaluate!`](@ref), [`Evaluation`](@ref),
-[`*(::Evaluation, ::Sequence)`](@ref) and [`(::Evaluation)(::Sequence)`](@ref).
+[`*(::Evaluation, ::AbstractSequence)`](@ref) and [`(::Evaluation)(::AbstractSequence)`](@ref).
 """
-(a::Sequence)(x, y...) = evaluate(a, (x, y...))
-(a::Sequence)(x) = evaluate(a, x)
+(a::AbstractSequence)(x, y...) = evaluate(a, (x, y...))
+(a::AbstractSequence)(x) = evaluate(a, x)
 
 """
     evaluate(a::Sequence, x)
diff --git a/src/sequence_spaces/special_operators/multiplication.jl b/src/sequence_spaces/special_operators/multiplication.jl
index c882bdd8..405c11ab 100644
--- a/src/sequence_spaces/special_operators/multiplication.jl
+++ b/src/sequence_spaces/special_operators/multiplication.jl
@@ -50,7 +50,7 @@ and [`^(::Sequence{<:SequenceSpace}, ::Int)`](@ref).
 
 Base.:+(ℳ::Multiplication) = Multiplication(+(sequence(ℳ)))
 Base.:-(ℳ::Multiplication) = Multiplication(-(sequence(ℳ)))
-Base.:^(ℳ::Multiplication, n::Int) = Multiplication(sequence(ℳ) ^ n)
+Base.:^(ℳ::Multiplication, n::Integer) = Multiplication(sequence(ℳ) ^ n)
 
 for f ∈ (:+, :-, :*)
     @eval begin
diff --git a/src/sequence_spaces/special_operators/shift.jl b/src/sequence_spaces/special_operators/shift.jl
index 499c1c82..43295a1e 100644
--- a/src/sequence_spaces/special_operators/shift.jl
+++ b/src/sequence_spaces/special_operators/shift.jl
@@ -39,29 +39,29 @@ value(𝒮::Shift) = 𝒮.value
 Base.:*(𝒮₁::Shift{<:Number}, 𝒮₂::Shift{<:Number}) = Shift(value(𝒮₁) + value(𝒮₂))
 Base.:*(𝒮₁::Shift{<:NTuple{N,Number}}, 𝒮₂::Shift{<:NTuple{N,Number}}) where {N} = Shift(map(+, value(𝒮₁), value(𝒮₂)))
 
-Base.:^(𝒮::Shift{<:Number}, n::Int) = Shift(value(𝒮) * n)
-Base.:^(𝒮::Shift{<:Tuple{Vararg{Number}}}, n::Int) = Shift(map(τᵢ -> *(τᵢ, n), value(𝒮)))
-Base.:^(𝒮::Shift{<:NTuple{N,Number}}, n::NTuple{N,Int}) where {N} = Shift(map(*, value(𝒮), n))
+Base.:^(𝒮::Shift{<:Number}, n::Integer) = Shift(value(𝒮) * n)
+Base.:^(𝒮::Shift{<:Tuple{Vararg{Number}}}, n::Integer) = Shift(map(τᵢ -> *(τᵢ, n), value(𝒮)))
+Base.:^(𝒮::Shift{<:NTuple{N,Number}}, n::NTuple{N,Integer}) where {N} = Shift(map(*, value(𝒮), n))
 
 """
-    *(𝒮::Shift, a::Sequence)
+    *(𝒮::Shift, a::AbstractSequence)
 
 Shift `a` by `value(𝒮)`; equivalent to `shift(a, value(𝒮))`.
 
-See also: [`(::Shift)(::Sequence)`](@ref), [`Shift`](@ref), [`shift`](@ref) and
+See also: [`(::Shift)(::AbstractSequence)`](@ref), [`Shift`](@ref), [`shift`](@ref) and
 [`shift!`](@ref).
 """
-Base.:*(𝒮::Shift, a::Sequence) = shift(a, value(𝒮))
+Base.:*(𝒮::Shift, a::AbstractSequence) = shift(a, value(𝒮))
 
 """
-    (𝒮::Shift)(a::Sequence)
+    (𝒮::Shift)(a::AbstractSequence)
 
 Shift `a` by `value(𝒮)`; equivalent to `shift(a, value(𝒮))`.
 
-See also: [`*(::Shift, ::Sequence)`](@ref), [`Shift`](@ref), [`shift`](@ref) and
+See also: [`*(::Shift, ::AbstractSequence)`](@ref), [`Shift`](@ref), [`shift`](@ref) and
 [`shift!`](@ref).
 """
-(𝒮::Shift)(a::Sequence) = *(𝒮, a)
+(𝒮::Shift)(a::AbstractSequence) = *(𝒮, a)
 
 """
     shift(a::Sequence, τ)