Summary Functional Analysis

Lecture 1
1.1 Linear spaces
Def 1.1 let X be a set and let K = R or C. Let X × X → X(x, y) 7→ x + y and K × X → X, (λx) 7→ λx
be maps . Then X is said to be a linear space over K if for all x, y, z ∈ X and λµ ∈ K the following
axioms are satisfied:
1. x + y = y + x
2. (x + y) + z = x + (y + z)
3. there exists 0 ∈ X such that x + 0 = x
4. there exists −x ∈ X such that x + (−x) = 0
5. λ(µx) = (λµ)x
6. 1x = x
7. λ(x + y) = λx + λy
8. (λ + µ)x = λx + µx
A subset V of a linear space X is called a linear subspace when it is a linear space itself with the given
operations.
Def 1.5 Let X be a linear space. The sum of two linear subspaces V, W ⊂ X is defined as
V + W = {x + y x ∈ V, y ∈ W }
The sum is called direct if V ∩ W = {0}.
Def 1.6 Let X be a linear space and let E ⊂ X be a set. The linear span of the set E is defined by
\
span(E) = {H ⊂ X : E ⊂ H , H linear subspace}

Prop 1.7 . The linear span of E is the unique linear subspace of X which contains E and is contained
in every linear subspace which contains E. In fact:
( n )
X
span(E) = λi ei n ∈ N, λi ∈ K, ei ∈ E, i = 1, . . . , n
i=1

Def 1.8 Let X be a linear space over K. A nonempty finite set M = {e1 , . . . , en } ⊂ X is called linearly
independent if with λ1 , . . . , λn ∈ K one has
n
X
λi ei = 0 =⇒ λ1 = · · · = λn = 0
i=1

A nonempty subset M ⊂ X is called linearly independent if every finite subset of M is linearly indepen-
dent. The dimension of X is defined by

0
 if X = {0}
dim X = n if X is spanned by n linearly independent vectors

∞ if X has an infinite linearly independent subset


A set of n linearly independent vectors which span X is called a basis for X.

1.2 Linear operators
Def 1.9 A relation T from X to Y is a set T ⊂ X × Y . If Y = X one speaks of a relation on the set X.
The domain and range of T are defined by
dom T = {x ∈ X : (x, y) ∈ T for some y ∈ Y }
ran T = {y ∈ Y : (x, y) ∈ T for some x ∈ X}


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,respectively. A relation is called a map T : X → Y if it satisfies the following property:

(x, y) ∈ T , (x, z) ∈ T =⇒ y = z

in which case one uses the notation y = T x. A map T : X → Y is called
1. injective if for every y ∈ Y there is at most one x ∈ X with y = T x
2. surjective if for every y ∈ Y there is at least one x ∈ X with y = T x
3. bijective if for every x ∈ X there is y ∈ Y with y = T x and for every y ∈ Y there is precisely one
x ∈ X with y = T x.
Def 1.10 Let X and Y be linear spaces over K. A map T : X → Y is called linear if T is everywhere
defined on X and if for all x, y ∈ X and λ ∈ K
1. T (x + y) = T x + T y
2. T (λx) = λ(T x)
The collection of all linear maps from X to Y is denoted by L(X, Y ).
Lemma 1.11 Let X and Y be linear spaces over K and let T : X → Y be a linear map. Then T is
bijective if and only if there exists a unique linear map S : Y → X such that ST = IX and T S = IY
Def 1.12 Let X be a linear space and let P : X → X be a linear map. Then P is called a projection if
P2 = P.
Lemma 1.13 A linear map P : X → X is a projection if and only if I − P is a projection. In this case:

ran P = ker (I − P ), ker P = ran (I − P ),

Moreover, X = ran P + ker P is a direct sum.
Def 1.14 Let X be a linear space and let V, W ⊂ X be linear subspaces. Then V and W are called
complementary if X = V + W is a direct sum. The subspaces induce corresponding linear projections
denoted by PV and PW with PV + PW = I
Def 1.15 Let X be a linear space over K and let T : X → X be a linear map. The point spectrum
σP (T ) of T is the set of all eigenvalues of T :

σP (T ) = {λ ∈ K : T x = λx for some x ̸= 0}

The geometric multiplicity of λ ∈ σP (T ) is the dimension of the corresponding eigenspace ker (T − λ)
Thm 1.16 Let X be a linear space and let T : X → X be a linear map. Eigenvectors corresponding to
different eigenvalues are linearly independent.

1.3 Quotient spaces of linear spaces
Def 1.17 A relation R on a set X is called an equivalence relation if
1. for each x ∈ X one has (x, x) ∈ R (reflexivity)
2. if (x, y) ∈ R, then (y, x) ∈ R (symmetry)
3. if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R (transitivity)
The statement (x, y) ∈ R is denoted by x y and the equivalence relation is denoted by . For x ∈ X the
equivalence class [x] of x is defined as

[x] = {y ∈ X : x y}

The set of all equivalence classes in X is denoted by X/ and the map π : X → X/ given by π(x) = [x]
is called the quotient map.
Thm 1.18 Let X be a set with an equivalence relation . Let x, y ∈ X, then one has the following
statements:
1. x ∈ [x]


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, 2. [x] = [y] ⇐⇒ x y
3. [x] ∩ [y] ̸= ∅ =⇒ [x] = [y]
S
4. X = [x], the disjoint union of all equivalence classes.
x∈X

Def 1.20 Let X be a linear space and let V ⊂ X be a linear subspace. Then V induces an equivalence
relation on X by
x y ⇐⇒ y − x ∈ V
The equivalence class to which x ∈ X belongs is denoted by x + V :
x + V = {y ∈ X y − x ∈ V }
The set of equivalence classes is denoted by X/V .
Prop 1.21 The space X/V provided with the following sum and scalar multiplication is a linear space
over K:
1. (x + V ) + (y + V ) = x + y + V , x, y ∈ X
2. λ(x + V ) = λx + V , x ∈ X, λ ∈ K
Def 1.22 Let X be a linear space and let V ⊂ X be a linear subspace. The quotient map pi : X → X/V
is defined by
π(x) = x + V, x ∈ X
Lemma 1.23 The map π is linear, surjective and ker π = V

1.4 Isomorphisms between linear spaces
Thm 1.24 Let X, Y be linear spaces and let V ⊂ X be a linear subspace.
1. Let T : X → Y be a linear map such that V ⊂ ker T . Then T induces a well-defined linear map
T̂ : X/V → Y, x + V 7→ T (x)
such that T = T̂ ◦ π
2. Let S : X/V → Y be a linear map. Then T = S ◦ π : X 7→ Y is a linear map with V ⊂ ker T
Corr 1.25 Let X, Y be linear spaces and let T : X → Y be a linear map. Then
T̂ : X/ker T → Y, x + ker T 7→ T (x)
is an injective linear map, so that X/ker T isomorphic to ran T . If in addition T is surjective, T̂ :
X/ker T → Y is an isomorphism of linear spaces.
Thm 1.26 Let X be a linear space with V ⊂ X a linear subspace. If dim X < ∞, then dim X/V < ∞
and dim X/V =dim X−dim V .
Corr 1.27 Let T : X → Y be a linear map with dim X < ∞. Then
dim ker T + dim ran T = dim X

1.5 Dual spaces of linear spaces
Def 1.28 Let X be a linear space over K. The dual space of X is defined as X ′ = L(X, K). The elements
of X ′ are called functionals on X.
Lemma 1.29 Let X be a finite-dimensional linear space. Then X ′ is a finite-dimensional linear space
and dim X ′ = dim X.
Def 1.30 Let X be a linear space over K. The second-dual space of X is defined as X ′′ = L(X ′ , K).
The natural map J : X → X ′′ is given by
J(x)(f ) = f (x), x ∈ X , f ∈ X′
Lemma 1.31 Let X be a finite-dimensional linear space. Then J : X → X ′′ is a bijection.
Def 1.32 Let X, Y be linear spaces and let T : X → Y be a linear operator. Then the conjugate operator
T × : Y ′ → X ′ is the linear operator defined by
(T × f )(x) = f (T x), f ∈Y′, x∈X
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, Lecture notes
A familiar example: Kn = {(x1 , . . . , xn ) | xi ∈ K}
Infinite-dimensional examples:
K∞ = {(x1 , x2 , . . . ) : xi ∈ K
F([a, b], K) = {f : [a, b] → K}
”Too large” for analysis purposes
Important examples:
∞
X
ℓp = {(x1 , x2 , . . . ) : xi ∈ K, |xi |p < ∞}, (p ≥ 1)
i=1

x = (1, 1/2, 1/3, . . . ) ̸∈ ℓ1
x = (1, 1/2, 1/3, . . . ) ∈ ℓ2
ℓ∞ = {(x1 , . . . ) : xi ∈ K, sup |xi | < ∞}
i∈N

C([a, b], K) = {f : [a, b] → K : f is continuous}
L(X, Y ) = {T : X → Y : T is linear}. If X = Y , we write L(X).
Example of projection: P : R2 → R2 , (x1 , x2 ) 7→ (0, x2 )
Lemma 1.13 proof:
Claim: (I − P ) is a projection
Proof: (I − P )2 = (I − P )(I − P ) = I − P − P + P 2 = I − 2P + P = I − P
Claim: ran P =ker (I − P ) and ker P =ran (I − P ).
Proof: x ∈ ran P ⇐⇒ x = P y for some y ∈ X ⇐⇒ P x = P 2 y = P y = x ⇐⇒ (I − P )x = 0 ⇐⇒
x ∈ ker (I − P ).
2nd claim follows from (I − P ) is a projection.
Claim: X = ker P + ran P
Proof: ⊃ is trivial
⊂: x = (I − P )x + P x for all x ∈ X
Direct sum: If x ∈ranP ∩ ker P , then, x = P y and P x = 0. Then x = P y = P 2 y = P x = 0 hence x = 0
Ex: X = {books with a single author}. x y ⇐⇒ x and y have the same author is an equivalence
relation on X.
Ex: on X = Z define the equivalence relation x y ⇐⇒ x − y is even.
[0] = {. . . , −2, 0, 2, . . . }, [1] = {. . . , −1, 1, . . . } hence Z/ = {[0], [1]}.
L(X, K) = {f : X → K : f is linear}




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