Teoremi e definizioni di Advanced Mathematics

THEOREMS AND DEFINITIONS

CHAPTER 1

DEF. 245) Given a vector space V, a positive function ||.|| is said to be a norm if:

• i) ||v|| = 0 ⇒ v = 0
• ii) ||v + w|| ≤ ||v|| + ||w|| ∀v,w ∈ V
• iii) ||λv|| = |λ| ||v|| ∀λ ∈ ℝ, ∀v ∈ V

DEF. 247) A vector space endowed with a norm ||.|| is called normed vector space.

LM. 248) Let (V, ||.||) be a normed vector space. The function d: V x V → ℝ defined by

d(v,w) = ||v - w|| ∀v,w ∈ V

is a distance with two additional properties:

• i) Translation invariance (d(v+x, w+x) = d(v,w))
• ii) Homogeneity (d(λv, λw) = |λ| d(v,w))

Not all distances in metric spaces are induced by norms.

DEF. 252) A normed vector space whose metric is complete is called a Banach space.

LM. 257) We have | | || || - || || ≤ || - ||

PROP. 259) A linear operator T: V₄ → V₂ between N.V.S. is continuous at a point v ∈ V₄ ⇔ it is continuous ∀ V₁.

DEF. 260) An operator T: V₄ → V₂ is called bounded if

∃K > 0 s.t.

||T(v)||₂ ≤ K ||v||₄ ∀v ∈ V₄

LM. 261) A linear operator T: V₄ → V₂ between N.V.S. is bounded ⇔ the image of any bounded subset of V₄ is a bounded subset of V₂.

TH. 263) A linear operator T: V₄ → V₂ between N.V.S. is continuous ⇔ it is bounded.

DEF.) The topological dual V* of a N.V.S. is the set of...

(OF ALL CONTINUOUS AND LINEAR FUNCTIONALS DEFINED ON V IT IS A VECTOR SUBSPACE OF THE ALGEBRAIC DUAL V'. THE DEFINITION IS ANALOGOUS FOR THE SET OF ALL CONTINUOUS AND LINEAR OPERATORS BETWEEN N.V.S. V_A AND V_B, B(V_A, V_B).

LM. 267)

LET T ∈ B(V_A, V₂), THE SET

{k ≥ 0 | ||T(v)||₂ ≤ k ||v||_A ∀ v ∈ V_A}

HAS A MINIMUM.

PROP. 269)

THE FUNCTIONAL ||·|| : B(V_A, V₂) → ℝ, DEFINED BY

||T|| = min{k ≥ 0 | ||T(v)||₂ ≤ k ||v||_A, ∀ v ∈ V_A}

IS A NORM.

PROP. 270)

LET T ∈ B(V_A, V₂), IT HOLDS:

||T|| = sup{||T(v)||₂, \( v ∈ V_A, ||v||_A = 1 \)}

= sup{||T(v)||₂, \( v ∈ B_VA \)}

= sup{||T(y)||₂, \( y ∈ S_VA \)}

TH. 273)

IF V₂ IS A BANACH SPACE, ALSO THE N.V.S. B(V_A, V₂) IS A BANACH SPACE.

PROPERTIES)

||T + S|| ≤ ||T|| + ||S||

||TS|| ≤ ||T|| ||S||

\( (S ∈ B(V_A, V₂), T ∈ B(V₂, V₃)) \)

CHAPTER 8

DEF.)

IN ITS GENERAL FORM, AN EQUATION CAN BE WRITTEN AS

^l(x) = y₀

DEF.)

THE INVERSE CORRESPONDENCE \( l^{-1} : Y \rightarrow 2^X \) IS DEFINED BY

\( l^{-1}(y) = \{x ∈ X | \phi(x) = y\} \) ∀ y ∈ Y

DEF.)

l IS WEAKLY INVERTIBLE AT y ∈ Y IF l^-1(y) IS NOT EMPTY (I.E., l IS SURJECTIVE W.R.T. & EXIST; y ∈ Y). l IS INVERTIBLE AT y ∈ Y IF l^-1(y) IS A SINGLETON (I.E., l IS BIJECTIVE W.R.T. & EXIST; y ∈ Y). IF THESE HOLD ∀ y ∈ Y, l IS (WEAKLY) INVERTIBLE. IT IS WELL POSED IF l^-1 ∈ B(X,Y) IS A CONTINUOUS FUNCTION.

TH. 337) A selfmap T:B(x)→B(x) is a β-contraction wrt the Blackwell

Sudnorm if it satisfies.

i) Monotonicity f(x) ≤ g(x) ⇒ T(f)(x) ≤ T(g)(x)

ii) Discounting Property: T(f+c) ≤ T(f)+βc ∀c≥0, β∈(0,1)

DEF.) The solutions of the Bellman equation are the fix points of the Bellman operator

T(f)(x):= sup {(x,y)+βf(y)} y∈P(x)

LM. 338) The Bellman operator is a β-contraction provided β∈(0,1) and is bounded.

PROP. 339) Then by B.C.T, the Bellman solution has a unique and globally attracting solution (because B(x) is a Banach space if X is a Banach space).

CHAPTER 11

DEF. 362) Let f:A⊆X→ℝ be a real valued function and

Let C be a subset of A. An element x̂∈C is a local (global) maximum of f on C if

f(x̂) ≥ f(x) ∀x∈C

The value f(x̂) is called maximum value of f on C.

PROP. 364) A point of maximum is strong (f(x̂)>f(x) ∀x∈C) ⇔ it is unique.

PROP. 365) Let g:B⊆ℝ→ℝ be a strictly increasing function with inf f∈B, the two problems of optimum

max_x sup f(x) x∈C

and

max_x sup (g∘f)(x) x∈C

are equivalent, that is, they have the same solution.

PROP. 366) Given f:A⊆X→ℝ, let C and C' be two any set

such that C⊆C'⊆A. We have

max_x∈C f(x) ≤ max_x∈C' f(x)

DEF.) The problem can thus be described as:

max_xt,d U(x_t;d)

sub (x_t,d) ∈ X ×D

DEF.) The dp problem is a special parametric optimization problem. Consider the feasibility correspondence

C(x₀) = {(x_t,d) ∈ X ×D |x_t+1 = ℓ(x_t,d_t) and d_t(x) ∀x∈X₀ given }

then the problem can be written as

max_xt,d U(x_t;d)

sub (x_t,d) ∈ C(x₀)

DEF.) The extended value function V: X→ℝ is given by

V(x₀) = sup_{(xt,d)∈C(x0)} U(x_t;d)

DEF.) The problem dp is uniquely controlled if there is a unique d_t that moves the system from state x to a certain state x_t+1. Formally, ∀x∈X

ℓ(x;d_t) = ℓ(x_t;d' ) ⇒ d_t=d' t.d.d∈ d(x)

There is a function g: X×X→D s.t.

ℓ(x,g(x;y))=y ∀x,y∈X

If the problem is not uniquely controlled, this function is a correspondence.

DEF.) The reduced form of a problem dp is given by

i) A space X of system states, who play the role of decision variables

ii) A dynamic constraint correspondence P: X→2^X that connects the current state x_t with the states x_b that can be reached after x_t.

iii) An intertemporal index Φ: Ω ⊂ X → ℝ given by

Φ(x) = ∑_k=0^∞β^kφ(x_k,x_k+1)

GOING BACK TO PARAMETRIC MAXIMIZATION PROBLEMS, IF WE CAN ALSO CHOOSE THE PARAMETER, WE HAVE

MAX f(x;θ)x,θ SUB x ∈ f(θ), θ ∈ Θ

OR

MAX f(x;θ)x,θ SUB (x,θ) ∈ GFL

THEN, BY PROP. L 41

SUP_(x;θ)∈GFL f(x;θ) = SUP_θ∈Θ (SUP_x∈f(θ) f(x;θ)) = SUP_θ∈Θ V(θ)

DEF.) IN DYNAMIC PROGRAMMING, WE USE TWO STEPS:

1. WE PARAMETERIZE THE PROBLEM BY TAKING X₀ AND τ AS PARAMETERS
2. WE DECOMPOSE THE PROBLEM IN TWO STAGES

DEF.) WE WANT TWO ASSUMPTIONS TO HOLD:

1. ∀x₀∈X, THE SET

(x₀) = {x∈x^∞ | x_n+1∈ β(x_n) ∀n≥0, x₀∈X GIVEN }

'FEASIBLE PATHS STARTING FROM X₀

IS NONEMPTY.

3. ∀x₀∈X, V(x₀)<+∞

⇒ V: X→ℝ IS DEFINED ∀ x₀∈X. ASS. 2 REQUIRES THAT

THE SERIES ϕ(x) = ∑_n=0^∞ βⁿ φ(x_n, x_n+1) CONVERGES ∀ x∈(x₀)

LM. 473) IF x₀∈X, ∃N_x₀>0 |

ϕ(x_n, x_n+1) ≤ N_x₀ ∀ x∈(x₀), ∀n≥0

THEN ASS. 2 HOLDS.

LM. 474) IF ϕ GFL→ℝ IS BOUNDED, THEN V: X→ℝ IS BOUNDED.

GIVEN A PATH X = {x₀, x₁, ....} ∈ x^∞, THE SHIFTED PATH IS

SX = {x₁, x₂, ....]

THE SHIFT OPERATOR S: x^∞→x^∞ DEFINED BY

S(x) = lim X

SHIFTS THE PATH ONE PERIOD AWAY.

TH. 475) THE VALUE FUNCTION V: X→ℝ OF PROBLEM RF SATISFIES THE BELLMAN EQUATION. THAT IS