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^-1(A ∩ B) = f^-1(A) ∩ f^-1(B)
• f(A) ⊆ f(A ∪ B) = f(A) ∪ f(B) (strict inclusion)
• 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

Σ = ∅, X, U∈S(A ∈ (σ -algebra) if:

• (i) ∅ ∈ ℓ ℓ ℓ
• (ii) Y ∈ ℓ ℓ ℓ, E ∈ ℓ ℓ ℓ (iii) yE ∈ ℓ ℓ ℓ
• (iii) Y E ⇌ ⊆ Σ ⇌ Σ U E ∈ ℓ ℓ ℓ

A B ∈ ℓ ℓ ℓ

METRIC SPACES

(X, d) METRIC SPACE, where d is a distance: X × X → [0, +∞), it is at.

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) < r }

x₀ is:

• A (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 ∀r>0: B_r(x₀) ∩ A ≠ 0 -> either interior/boundary
• 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 = { x₀ }

Isolated => adherence, but not accumulation

Accumulation => adherence, but not isolated

 interior

ext(A) exterior

A boundary

 closure (∪ adherence points)

A' derived (∪ limit points)

Â, ext(A), ∂A -> Partition of A

 = A ∪ Q A =  ∪ ∂A A' =  \  ∣ isolated points ∈

A net A is open if every x∈A is an interior point. A is closed if it complements A^c 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.

A˚ is the union of all open sets contained in A. => it’s the largest open subset in A.

A open => A= A˚_˚ A˚=A A∩∂A=0

A closed => A=A̅ A̅⊂A

A₁,...,A_N open => _i=1∧_N A_j open ⎫ _open -> wrt finite intersect and _j∈N U_j∈N A_j open ⎬ arbitrary unions

A₁,...,A_N closed => _j=1∧_N A_j closed ⎫ _closed -> wrt finite unions and _j∈N ∩_j A_j closed ⎬ arbitrary intersections

limit of req. in metric spaces

∃x* req. in (X, d) x∈X: x_n→x* n→∞ if ∀ε>0, ∃n=π(ε): n≥n̅ => d(x_n,x*) < ε

G unique

A ⊂ X closed ∀x∃x_n req. in A, X_n → x* => x*∈A

A closed A req. closed in metric spaces.

limit of f in metric spaces

y∈Y is the limit for x→x_o of f(x) if ∀ε≥0 ∃δ(x_oε) r.c d(x,x_o)∈S => d(y,f(x,y_o) < ε

CONTINUITY

f continuous in x_o if: x_o is accumul. point of X and f(x_o)=lim f(x) x→x_o x_o is isolated point

seq. conver.

f is seq. cont. in x_o if ∀x_n∃x: x_n→x_o => f(x_n)→f(x_o)

f continuous f seq. cont. in metric spaces

COMPLETENESS

Cauchy req.

∀x_n Cauchy req. in X if ∀ε≥0, ∃n̅=π(ε) ∈ N st. ∀n,μ≥π => d(x_nx_μ)∈ε

(X,d) is complete if any Cauchy seq. in X converges to a limit

(x_n is a conv. req.) x_n t←x_n+1 Cauchy (