Schemi riassuntivi di Analisi reale e funzionale

Real analysis

Set theory

Functions

f^-1(A^c) = (f^-1(A))^c

f^-1(A ∪ B) = f^-1(A) ∪ f^-1(B)

f(A ∪ B) ⊆ f(A) ∪ f(B)

f(A ∩ B) ⊆ f(A) ∩ f(B) (strict inclusion)

f(A ∩ B) = f(A) ∩ f(B)

f^-1(A ∩ B) = f^-1(A) ∩ f^-1(B)

f^-1(A ∪ B) = f^-1(A) ∪ f^-1(B)

Injection: f(x) ≠ f(x') whenever x ≠ x' ⇒ ∃ f^-1

Surjection: If range (f) = Y or f(X) = Y, f: X → Y

Measure theory

σ-algebra

- ∅, ∈℘() is a σ-algebra if:

• (i) ∅ ∈
• (ii) ∀E∈, E^c∈
• (iii) ∀{E_n}_n ∈, ⋃E_n ∈ ⇒ holds that: ∀{E_n} ⊆ ⇒ ⋂E_n ∈, A ∩ B ∈

Metric spaces

(X,d) metric space, where d is a distance: XxX → [0,+∞), it is ∀t.d(x,y) ≥ 0 + d(x,y) = 0 ↔ x = y, d(x,y) = d(y,x), d(x,y) = d(x,z)+d(z,y),

open ball of center x₀ and radius r: B_r(x₀), ∃ x ∈ X. d(x,x₀) x₀ is: (A ⊆ X)

Interior point of A if ∃ r > 0: B_r(x₀) ⊆ A

Exterior point of A if ∃ r > 0: B_r(x₀) ⊆ A^c

Boundary point of A if x₀ is neither interior nor exterior

Adherence point of A if ∀ r>0 : B_r(x₀) ∩ A ≠ ∅ → either interior/boundary

Accumulation/limit/cluster point of A if ∀r>0: ∃x∈(B_r(x₀) ∩ A) ∖ {x₀}

Isolated point of A if x₀∈A and ∃ r>0: B_r(x₀) ∩ A = 3 x₀

Isolated ⇒ adherence, but not accumulation

Accumulation ⇒ adherence, but not isolated

Ā interior

ext(A) exterior

∂A boundary

Ā closure (set adherence points)

A' derived (set limit points)

Ā, ext(A), ∂A = Partition of A

Ā = A ∪ ∂A = A ∪ ext(A)

A' = Ā ∖ A (isolated points)

Real analysis

Set theory

Functions

f^-1(A^C) = [f^-1(A)]^C

f^-1(A ∪ B) = f^-1(A) ∪ f^-1(B)

f^-1(A ∩ B) = f^-1(A) ∩ f^-1(B)

A ⊆ B ⇒ f^-1(A) ⊆ f^-1(B) (strict inclusion)

f(A ∩ B) ⊆ f(A) ∩ f(B)

f(A ∪ B) = f(A) ∪ f(B)

Injection: f(x) ≠ f(x') whenever x ≠ x' ⇒ ∃f^-1

Surjection: If range (f) = Y or f(X) = Y ⇒ f : X → Y

Measure theory

σ-algebra

∅≠∅, M⊆P(Ω) is a σ-algebra if:

• (i) ∅ ∈ M
• (ii) ∀ E ∈ M, E^c ∈ M
• (iii) ∀ E_i ∈ M, ⋃E_i ∈ M=> holds that: ∀E_i ⊆ M ⇒ ⋂E_i ∈ M, ∀A,B ∈ M

Metric spaces

(X,d) metric space, where d is a distance: X×X → [0,+∞), it is st. d(x,y)≥0 + d(x,y)=0 x=y, d(x,y)=d(y,x), d(x,y)≤d(x,z)+d(z,y),

open ball of center x₀ and radius r: B_r(x₀) = {x ∈ X : d(x,x₀) x₀ is: (A⊆X)

Interior point of A if ∃r>0: B_r(x₀) ⊆ A

Exterior point of A if ∃r>0: B_r(x₀) ⊆ A^c

Boundary point of A if X ≠ interior nor exterior

Adherence point of A if ∀r>0: B_r(x₀) ∩ A ≠ ∅ → either interior/boundary

Accumulation/limit/cluster point of A if ∀r>0: ∃x ∈ (B_r(x₀) ∩ A) \ {x₀}

Isolated point of A if x₀∈A and ∃r>0: B_r(x₀) ∩ A = 3x₀

Isolated ⇒ adherence, but not accumulation

Accumulation ⇒ adherence, but not isolated

 interior

ext(A) exterior

∂A boundary

Ā closure (per adherence points)

A^' derived (per limit points)

Ā, ext(A), ∂A = Partition of A

Ā̅ = A ∪ ∂A = A ∪ ∂A

A^' = Ā \ A \ isolated points

A set A is open if every x ∈ A is an interior point.

A is closed if X \ A is open.

A̅ is the intersection of all closed sets containing A (closure ≅ point in A or near A) ⇒ it's the smallest closed subset of X containing A.

Å is the union of all open sets contained in A. ⇒ it is the largest open subset in A.

A open ⇔ A̅ = Å ⇔ A ∩ ∂A = 0

A closed ⇔ A = A̅ ⇔ ∂A ⊆ A

• A₁, ..., A_n open ⇒ ⋂ (from j=1 to n) A_j open
• ∀A_j open ⇒ ⋃ (j ∈ J) A_j open
• { Open → wrt finite intersection and arbitrary unions }
• A₁, ..., A_n closed ⇒ ⋃ (from j=1 to n) A_j closed
• ∀A_j closed ⇒ ⋂ (j ∈ J) A_j closed
• { Closed → wrt finite union and arbitrary intersections }

Limit of sequences in metric spaces

∃! x_n seq. in (X, d) x_n ∈ X

x_n → x^*

x_n → x^* if ∀ε > 0, ∃ n̅= n̅(ε): ∀n > n