Algebraic dual space Dual space

1 algebraic dual space

1.1 finite-dimensional case
1.2 infinite-dimensional case
1.3 bilinear products , dual spaces
1.4 injection double-dual
1.5 transpose of linear map
1.6 quotient spaces , annihilators

algebraic dual space

given vector space v on field f, (algebraic) dual space v (alternatively denoted




v

∨




{\displaystyle v^{\vee }}

or




v
′



{\displaystyle v }

) defined set of linear maps φ: v → f (linear functionals). since linear maps vector space homomorphisms, dual space denoted hom(v, f). dual space v becomes vector space on f when equipped addition , scalar multiplication satisfying:

(
φ
+
ψ
)
(
x
)
=
φ
(
x
)
+
ψ
(
x
)






(
a
φ
)
(
x
)
=
a

(
φ
(
x
)
)







{\displaystyle {\begin{aligned}&(\varphi +\psi )(x)=\varphi (x)+\psi (x)\\&(a\varphi )(x)=a\left(\varphi (x)\right)\end{aligned}}}

for φ , ψ ∈ v, x ∈ v, , ∈ f. elements of algebraic dual space v called covectors or one-forms.

the pairing of functional φ in dual space v , element x of v denoted bracket: φ(x) = [x,φ] or φ(x) = ⟨φ,x⟩. pairing defines nondegenerate bilinear mapping ⟨·,·⟩ : v × v → f called natural pairing.

finite-dimensional case

if v finite-dimensional, v has same dimension v. given basis {e1, ..., en} in v, possible construct specific basis in v, called dual basis. dual basis set {e, ..., e} of linear functionals on v, defined relation

e


i


(

c

1




e


1


+
⋯
+

c

n




e


n


)
=

c

i


,

i
=
1
,
…
,
n


{\displaystyle \mathbf {e} ^{i}(c^{1}\mathbf {e} _{1}+\cdots +c^{n}\mathbf {e} _{n})=c^{i},\quad i=1,\ldots ,n}

for choice of coefficients c ∈ f. in particular, letting in turn each 1 of coefficients equal 1 , other coefficients zero, gives system of equations

e


i


(


e


j


)
=

δ

j


i




{\displaystyle \mathbf {e} ^{i}(\mathbf {e} _{j})=\delta _{j}^{i}}

where




δ

j


i




{\displaystyle \delta _{j}^{i}}

kronecker delta symbol. example, if v r, , basis chosen {e1 = (1, 0), e2 = (0, 1)}, e , e one-forms (functions map vector scalar) such e(e1) = 1, e(e2) = 0, e(e1) = 0, , e(e2) = 1. (note: superscript here index, not exponent).

in particular, if interpret r space of columns of n real numbers, dual space typically written space of rows of n real numbers. such row acts on r linear functional ordinary matrix multiplication. 1 way see functional maps every n-vector x real number y. then, seeing functional matrix m, , x, y n × 1 matrix , 1 × 1 matrix (trivially, real number) respectively, if have mx = y, then, dimension reasons, m must 1 × n matrix, i.e., m must row vector.

if v consists of space of geometrical vectors in plane, level curves of element of v form family of parallel lines in v, because range 1-dimensional, every point in range multiple of 1 nonzero element. element of v can intuitively thought of particular family of parallel lines covering plane. compute value of functional on given vector, 1 needs determine of lines vector lies on. or, informally, 1 counts how many lines vector crosses. more generally, if v vector space of dimension, level sets of linear functional in v parallel hyperplanes in v, , action of linear functional on vector can visualized in terms of these hyperplanes.

infinite-dimensional case

if v not finite-dimensional has basis eα indexed infinite set a, same construction in finite-dimensional case yields linearly independent elements e (α ∈ a) of dual space, not form basis.

consider, instance, space r, elements sequences of real numbers contain finitely many non-zero entries, has basis indexed natural numbers n: ∈ n, ei sequence consisting of zeroes except in ith position, 1. dual space of r (isomorphic to) r, space of sequences of real numbers: such sequence (an) applied element (xn) of r give number ∑anxn, finite sum because there finitely many nonzero xn. dimension of r countably infinite, whereas r not have countable basis.

this observation generalizes infinite-dimensional vector space v on field f: choice of basis {eα : α ∈ a} identifies v space (f)0 of functions f : → f such fα = f(α) nonzero finitely many α ∈ a, such function f identified vector

∑

α
∈
a



f

α




e


α




{\displaystyle \sum _{\alpha \in a}f_{\alpha }\mathbf {e} _{\alpha }}

in v (the sum finite assumption on f, , v ∈ v may written in way definition of basis).

the dual space of v may identified space f of functions f: linear functional t on v uniquely determined values θα = t(eα) takes on basis of v, , function θ : → f (with θ(α) = θα) defines linear functional t on v by

t


(



∑

α
∈
a



f

α




e


α




)


=

∑

α
∈
a



f

α


t
(

e

α


)
=

∑

α
∈
a



f

α



θ

α


.


{\displaystyle t{\biggl (}\sum _{\alpha \in a}f_{\alpha }\mathbf {e} _{\alpha }{\biggr )}=\sum _{\alpha \in a}f_{\alpha }t(e_{\alpha })=\sum _{\alpha \in a}f_{\alpha }\theta _{\alpha }.}

again sum finite because fα nonzero finitely many α.

note (f)0 may identified (essentially definition) direct sum of infinitely many copies of f (viewed 1-dimensional vector space on itself) indexed a, i.e., there linear isomorphisms

v
≅
(

f

a



)

0


≅

⨁

α
∈
a



f

.


{\displaystyle v\cong (f^{a})_{0}\cong \bigoplus _{\alpha \in a}{f}.}

on other hand, f (again definition), direct product of infinitely many copies of f indexed a, , identification

v

∗


≅


(



⨁

α
∈
a


f



)



∗


≅

∏

α
∈
a



f

∗


≅

∏

α
∈
a


f
≅

f

a




{\displaystyle v^{*}\cong {\biggl (}\bigoplus _{\alpha \in a}f{\biggr )}^{*}\cong \prod _{\alpha \in a}f^{*}\cong \prod _{\alpha \in a}f\cong f^{a}}

is special case of general result relating direct sums (of modules) direct products.

thus if basis infinite, algebraic dual space of larger dimension (as cardinal number) original vector space. in contrast case of continuous dual space, discussed below, may isomorphic original vector space if latter infinite-dimensional.

bilinear products , dual spaces

if v finite-dimensional, v isomorphic v. there in general no natural isomorphism between these 2 spaces. bilinear form ⟨·,·⟩ on v gives mapping of v dual space via

v
↦
⟨
v
,
⋅
⟩


{\displaystyle v\mapsto \langle v,\cdot \rangle }

where right hand side defined functional on v taking each w ∈ v ⟨v,w⟩. in other words, bilinear form determines linear mapping

Φ

⟨
⋅
,
⋅
⟩


:
v
→

v

∗




{\displaystyle \phi _{\langle \cdot ,\cdot \rangle }:v\to v^{*}}

defined by

[

Φ

⟨
⋅
,
⋅
⟩


(
v
)
,
w
]
=
⟨
v
,
w
⟩
.


{\displaystyle [\phi _{\langle \cdot ,\cdot \rangle }(v),w]=\langle v,w\rangle .}

if bilinear form nondegenerate, isomorphism onto subspace of v. if v finite-dimensional, isomorphism onto of v. conversely, isomorphism



Φ


{\displaystyle \phi }

v subspace of v (resp., of v if v finite dimensional) defines unique nondegenerate bilinear form



⟨
⋅
,
⋅

⟩

Φ




{\displaystyle \langle \cdot ,\cdot \rangle _{\phi }}

on v by

⟨
v
,
w

⟩

Φ


=
(
Φ
(
v
)
)
(
w
)
=
[
Φ
(
v
)
,
w
]
.



{\displaystyle \langle v,w\rangle _{\phi }=(\phi (v))(w)=[\phi (v),w].\,}

thus there one-to-one correspondence between isomorphisms of v subspaces of (resp., of) v , nondegenerate bilinear forms on v.

if vector space v on complex field, more natural consider sesquilinear forms instead of bilinear forms. in case, given sesquilinear form ⟨·,·⟩ determines isomorphism of v complex conjugate of dual space

Φ

⟨
⋅
,
⋅
⟩


:
v
→



v

∗


¯


.


{\displaystyle \phi _{\langle \cdot ,\cdot \rangle }:v\to {\overline {v^{*}}}.}

the conjugate space v can identified set of additive complex-valued functionals f: v → c such that

f
(
α
v
)
=


α
¯


f
(
v
)
.


{\displaystyle f(\alpha v)={\overline {\alpha }}f(v).}



injection double-dual

there natural homomorphism



Ψ


{\displaystyle \psi }





v


{\displaystyle v}

double dual




v

∗
∗


=
{
Φ
:

v

∗


→
f
:
Φ
 

l
i
n
e
a
r

}


{\displaystyle v^{**}=\{\phi :v^{*}\to f:\phi \ \mathrm {linear} \}}

, defined by



(
Ψ
(
v
)
)
(
φ
)
=
φ
(
v
)


{\displaystyle (\psi (v))(\varphi )=\varphi (v)}





v
∈
v
,
φ
∈

v

∗




{\displaystyle v\in v,\varphi \in v^{*}}

. in other words, if





e
v


v


:

v

∗


→
f


{\displaystyle \mathrm {ev} _{v}:v^{*}\to f}

evaluation map defined



φ
↦
φ
(
v
)


{\displaystyle \varphi \mapsto \varphi (v)}

, define



Ψ
:
v
→

v

∗
∗




{\displaystyle \psi :v\to v^{**}}

map



v
↦


e
v


v




{\displaystyle v\mapsto \mathrm {ev} _{v}}

. map



Ψ


{\displaystyle \psi }

injective; isomorphism if , if



v


{\displaystyle v}

finite-dimensional. indeed, isomorphism of finite-dimensional vector space double dual archetypal example of natural isomorphism. note infinite-dimensional hilbert spaces not counterexample this, isomorphic continuous duals, not algebraic duals.

transpose of linear map

if f : v → w linear map, transpose (or dual) f : w → v defined by

f

∗


(
φ
)
=
φ
∘
f



{\displaystyle f^{*}(\varphi )=\varphi \circ f\,}

for every φ ∈ w. resulting functional f(φ) in v called pullback of φ along f.

the following identity holds φ ∈ w , v ∈ v:

[

f

∗


(
φ
)
,

v
]
=
[
φ
,

f
(
v
)
]
,


{\displaystyle [f^{*}(\varphi ),\,v]=[\varphi ,\,f(v)],}

where bracket [·,·] on left natural pairing of v dual space, , on right natural pairing of w dual. identity characterizes transpose, , formally similar definition of adjoint.

the assignment f ↦ f produces injective linear map between space of linear operators v w , space of linear operators w v; homomorphism isomorphism if , if w finite-dimensional. if v = w space of linear maps algebra under composition of maps, , assignment antihomomorphism of algebras, meaning (fg) = gf. in language of category theory, taking dual of vector spaces , transpose of linear maps therefore contravariant functor category of vector spaces on f itself. note 1 can identify (f) f using natural injection double dual.

if linear map f represented matrix respect 2 bases of v , w, f represented transpose matrix respect dual bases of w , v, hence name. alternatively, f represented acting on left on column vectors, f represented same matrix acting on right on row vectors. these points of view related canonical inner product on r, identifies space of column vectors dual space of row vectors.

quotient spaces , annihilators

let s subset of v. annihilator of s in v, denoted here s, collection of linear functionals f ∈ v such [f, s] = 0 s ∈ s. is, s consists of linear functionals f : v → f such restriction s vanishes: f|s = 0.

the annihilator of subset vector space. in particular, ∅ = v of v (vacuously), whereas v = 0 0 subspace. furthermore, assignment of annihilator subset of v reverses inclusions, if s ⊂ t ⊂ v, then

0
⊂

t

o


⊂

s

o


⊂

v

∗


.


{\displaystyle 0\subset t^{o}\subset s^{o}\subset v^{*}.}

moreover, if , b 2 subsets of v, then

(
a
∩
b

)

o


⊇

a

o


+

b

o


,


{\displaystyle (a\cap b)^{o}\supseteq a^{o}+b^{o},}

and equality holds provided v finite-dimensional. if ai family of subsets of v indexed belonging index set i, then

(

⋃

i
∈
i



a

i


)


o


=

⋂

i
∈
i



a

i


o


.


{\displaystyle \left(\bigcup _{i\in i}a_{i}\right)^{o}=\bigcap _{i\in i}a_{i}^{o}.}

in particular if , b subspaces of v, follows that

(
a
+
b

)

o


=

a

o


∩

b

o


.


{\displaystyle (a+b)^{o}=a^{o}\cap b^{o}.}

if v finite-dimensional, , w vector subspace, then

w

o
o


=
w


{\displaystyle w^{oo}=w}

after identifying w image in second dual space under double duality isomorphism v ≈ v. thus, in particular, forming annihilator galois connection on lattice of subsets of finite-dimensional vector space.

if w subspace of v quotient space v/w vector space in own right, , has dual. first isomorphism theorem, functional f : v → f factors through v/w if , if w in kernel of f. there isomorphism

(
v

/

w

)

∗


≅

w

o


.


{\displaystyle (v/w)^{*}\cong w^{o}.}

as particular consequence, if v direct sum of 2 subspaces , b, v direct sum of , b.