Analysis in Several Real Variables

sorted by chaters of keilthy's notes for referencing convenience

Ch. 0: Thm: Monotonic and bounded sequences… (0.17)

Converge 

Ch. 0: Bolzano-Weierstrass Theorem (0.18)

Let {x_n} be a bounded sequence. Then there exists a subsequence {x_jk}_k=1^infinity , with j_k < j_k+1 that converges to a finite limit.

Ch. 1: Def: Euclidean Norm and inner product (1.2)

|| \vec(x) || = \sqrt(x₁² + x₂² + … + x_n² ) and ⟨x, y⟩ = x₁y₁ + x₂y₂ + … + x_ny_n

Ch. 1: Convergence of sequence wrt Euclidean norm (1.3)

A sequence {\vec(x)_n } of points in Rⁿ is a said to converge to a point p \in Rⁿ if for any epsilon > 0, there exists an integer N > 0 sucj that || \vec(x)_k - \vec(p) || < \epsilon for all k >= N. 

Ch. 1: Lemma: Components are … (1.4)

bounded by norm; i.e. if \vec(x) and \vec(y) \in Rⁿ then |x_i - y_i | =< || \vec(x) - \vec(y) || for all 1 =< i =< n

Ch.1: Lemma: Convergence and components (1.5)

Sequences converge iff their components do.

Ch. 1: Bolzano-Weierstrauss Theorem p.2 (1.6)

Every bounded sequences of points in Rⁿ has a convergent subsequence.

Ch. 1: Def: Cauchy Sequence (1.7)

A sequence {\vec(x)} in Rⁿ is called Cauchy if for every epsilon > 0 there exists N > 0 such that || \vec(x)_k - \vec(x)_l || < epsilon for every k, l >= N. (note every such cauchy seq. is bounded)

Ch. 1: Prop: The Cauchy-Schwarz Inequality (1.8)

Let \vec(x), \vec(y) \in Rⁿ. Then |⟨⃗x, ⃗y⟩| ≤ ∥⃗x∥∥⃗y∥ with equality if and only if ⃗x is a scalar multiple of ⃗y or vice versa.

Ch. 1: Corollary: Triangle Equality (pos and neg) (1.9-1.10)

Let ⃗x, ⃗y ∈ R^m. Then ∥⃗x + ⃗y∥ ≤ ∥⃗x∥ + ∥⃗y∥, and ∥⃗x − ⃗y∥ ≥ ∥⃗x∥ − ∥⃗y∥.

Ch. 1: Theorem: About cauchy sequences.. (1.12)

A sequence converges in R^m if and only if it is Cauchy.

Ch.2: Def: Open and Close Balls (2.1 & 2.8)

Given a point ⃗p ∈ R^m and a positive real number r > 0, the open ball of radius r, centred at ⃗p is B(⃗p, r) := {⃗x ∈ R^m | ∥⃗x − ⃗p∥ < r} and B(⃗p, r) (with a bar over) := {⃗x ∈ R^m | ∥⃗x − ⃗p∥ ≤ r}. 2.8 defines it for a subset X \in R^m .

Ch. 2: Def: Open (2.2)

A subset U ⊂ R^m is called open if for every ⃗x ∈ U, there exists a real number r_x > 0 such that B(⃗x, r_x) ⊂ U (define empty set to be open).

Ch. 2: Lemma: U open in R^m and X subset of R^m (2.9)

then U ∩ X is open in X

Ch. 2: Corollary: Open balls…

Are open (bless)

Ch. 2: Def.: Topology on a set X (2.13)

is a collection τ of subsets of X, called open sets in X, such that

i) Both X and the empty set are open: ∅, X ∈ τ

ii) The union of any collection of open sets is open

iii) The intersection of any finite collection of open sets is open

Ch. 2: Prop.: An easy topology on X⊂ R^m (2.14)

is the collection of open sets in X

Ch. 2: Prop: for W ⊂ X ⊂ R^m. Then W is open in X if and only if (2.17)

there exists U open in R^m such that W = U ∩ X

Ch.2: Def.: Closed sets (2.1)

Let X ⊂ R^m. We call a subset F ⊂ X closed in X if and only if X \ F is open in X. (closed balls are closed) ((bless))

Ch. 2: Prop.: Let X ⊂ R^m. The collection of closed sets in X satisfies the following properties: (2.23)

i) The empty set ∅ and X are closed in X

ii) The intersection of any collection of closed sets in X is closed in X

iii) The union of any finite collection of closed sets in X is closed in X

Ch.2 : Lemma: Convergent sequences in open sets (2.24)

converge to p iff any open set U containing p, there exists N > 0 such that ⃗xn ∈ U for every n ≥ N

Ch. 2: Lemma: Convergent sequences in closed sets (2.25)

converge in the closed set

Ch. 3: Def. : Continuous functions φ : X → Y,  X ⊂ R^m and Y ⊂ Rⁿ (3.1)

is called continuous at a point ⃗p ∈ X if for every ε > 0, there exists δ > 0 such that ∥φ(⃗x) − φ(⃗p)∥ < ε for every ⃗x ∈ X with ∥⃗x − ⃗p∥ < δ.

φ : X → Y is continuous on X if φ is continuous at every point ⃗p of X.

Ch. 3: Prop: Composition of continuous functions (3.2)

are continuous

Ch. 3: Prop.: Continuous functions and limits (3.3 & 3.4)

“Commute”; let φ : X → Y continuous at a point, Suppose {⃗x_k} ⊂ X converges to p, then {φ(⃗x_k)} converges to φ(⃗p) in Y. In fact a function is continous everywhere iff for all convergent sequences in X the above is true

Ch. 3: Lemma: Coordinate maps? (3.6)

also continuous!

Ch. 3: Prop: A function is continuous iff (3.7)

its components are

Ch. 3: Lemma: Sums and products (3.8)

are continuous

Ch. 3: Prop.: A function φ : X → Y is continuous iff (3.12 & 3.13)

φ^-1 (V ) is open in X for every V open in Y, similarly for closed sets.

Ch. 3: Corollary: Let f : X → R be continuous, and let a, b, c ∈ R with a < b. examples of open and closed sets (3.14)

{⃗x ∈ X | f(⃗x) > c}, {⃗x ∈ X | f(⃗x) < c}, {⃗x ∈ X | a < f(⃗x) < b}, {⃗x ∈ X | f(⃗x) ̸= c}, are all open in X

{⃗x ∈ X | f(⃗x) ≥ c}, {⃗x ∈ X | f(⃗x) ≤ c}, {⃗x ∈ X | a ≤ f(⃗x) ≤ b}, {⃗x ∈ X | f(⃗x) = c}, are closed in X

Ch. 3: Extreme Value Theorem (3.17)

Continuous functions on closed and bounded sets achieve their extrema

Ch.3: Def.: uniformly continuous functions (3.18)

if for any ε > 0, there exists δ > 0 such that ∥φ(⃗u) − φ(⃗v)∥ < ε for all ⃗u, ⃗v ∈ X such that ∥⃗u − ⃗v∥ < δ. Note that δ cannot depend on ⃗u or ⃗v.

Ch. 3: Theorem: Continuous functions on closed and bounded sets (3.20

are uniformly continuous

Ch. 4: Def.: Limit Point (4.1)

A point if for all ε > 0 B(⃗p, ε) ∩ X contains a point other than ⃗p, i.e. there exists ⃗x ∈ X such that 0 < ∥⃗x − ⃗p∥ < ε.

Ch. 4: Prop.: Closedness related to limit points (4.4)

A set X ∈ R^m is closed if and only if X contains all its limit points.

Ch. 4: Def.: Limit (4.5)

Let X ⊂ R^m and let ⃗p ∈ R m be a limit point of X. Suppose we have a function ϕ : X → Rⁿ and a point ⃗v ∈ Rⁿ. The point ⃗v is called the limit of ϕ(⃗x) as ⃗x tends to ⃗p if and only if for all ε > 0, there exists δ > 0 such that ∥ϕ(⃗x) − ⃗v∥ < ε for all ⃗x ∈ X satisfying 0 < ∥⃗x − ⃗p∥ < δ. We write lim ⃗x→⃗p ϕ(⃗x) = ⃗v.

Ch. 4: Prop.: Limits and components (4.7)

Limits can be computed in terms of components

Ch. 4: Prop.: Linear combinations of limits are.. (4.8)

limits of linear combinations

Ch. 4: Lemma: Limit points and continuity (4.9)

Let X ⊂ R^m have a limit point ⃗p contained within X. Then a function ϕ : X → Rⁿ is continuous at ⃗p if and only if lim⃗x→⃗p ϕ(⃗x) = ϕ(⃗p)

Ch. 4: Lemma: Limits and continuous functions (4.10)

Limits commute with continuous functions

Ch. 5: Rolle’s Theorem (5.9)

Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Suppose further that f(a) = f(b). Then there exists c ∈ (a, b) such that f ′ (c) = 0.

Ch. 5: Mean Value Theorem (5.10)

Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Then there exists c ∈ (a, b) such that (f(b) − f(a)) / (b − a) = f ′ (c).

Ch. 5: Def.: Real Analytic Function (5.15)

We call a function f : (a, b) → R real analytic on (a, b) if the n^th derivative f⁽ⁿ⁾ (x) exists for all x ∈ (a, b) and n ≥ 0, and for every x ∈ (a, b) there exists δ > 0 such that f(x + h) = \sum^∞_{n=0} f⁽ⁿ⁾(x) hⁿ / n!, for all |h| < δ. This power series is called the Taylor series of f around x.

Ch. 5: Taylor’s Theorem (5.17)

Let x₀, h ∈ R and let f be a k-times differentiable function defined on an open interval containing both x₀ and x₀ + h. Then f(x₀ + h) = \sum^{k−1}_{n=0} f⁽ⁿ⁾(x₀) hⁿ / n! + f^(k) (x0 + θh) h^k / k! for some θ ∈ (0, 1). (“Taylor series mostly work!”)

Ch. 6: Def.: Directional derivative and partial derivative (6.1)

Let X ⊂ R^m and let f : X → R be a function defined around a point ⃗x₀ ∈ X. Let ⃗v ∈ R m be a unit vector. The directional derivative of f (in the direction of ⃗v) at ⃗x₀ is the limit ∂⃗v_f(⃗x₀) := lim_h→0 ( f(⃗x₀ + h⃗v) − f(⃗x₀) )/ h assuming the limit exists. (where partial derivatives are defined as directional derivatives along the standard basis)

Ch. 6: Corollary: Bounded partial derivaitves imply (6.10)

continuity

Ch. 6: Def.: Gradient of a function (6.12)

Let X be an open set in R^m, and let f : X → R be a function whose first order partial derivatives are defined at a point ⃗p ∈ X. The gradient of f at ⃗p is the element of R m given by (∇f)_p = (∂₁f(⃗p), ∂₂f(⃗p), . . . ∂_mf(⃗p))

Ch. 6: Def. : Jacobian (6.15)

Let X ⊂ R^m be open, and suppose φ : X → R n is a function with components φ(⃗x) = (f₁(⃗x), . . . , f_n(⃗x)). Suppose further that the partial derivatives ∂_jf_i exist at a point ⃗p ∈ X for each 1 ≤ i ≤ n and 1 ≤ j ≤ m. Define the Jacobian of φ at ⃗p to be the (n×m)-matrix with components ((J φ)_p )_i,j := ∂_jf_i(⃗p).

Ch. 6: Def.: “φ is differentiable at ⃗p with derivative T” (6.18)

Let X ⊂ R^m be an open set, and let φ : X → Rⁿ be a function. Let T : R^m → Rⁿ be a linear transformation, and let ⃗p ∈ X be a point. We say that φ is differentiable at ⃗p with derivative T if and only if lim _⃗x→⃗p ( φ(⃗x) − φ(⃗p) − T(⃗x − ⃗p) ) / ( ∥⃗x − ⃗p∥ ) = 0.

Ch. 6: Lemma:The derivative of a linear map is (6.19)

linear

Ch. 6: Lemma: Differentiable functions satisfy inequalities (6.20)

Let X ⊂ R^m be an open set, and let φ : X → R n be a function on X. Then φ is differentiable at a point ⃗p ∈ X, with derivative T if and only if for every ε > 0, there exists δ > 0 such that ∥φ(⃗x) − φ(⃗p) − T(⃗x − ⃗p)∥ ≤ ε∥⃗x − ⃗p∥ for all ⃗x ∈ X such that ∥⃗x − ⃗p∥ < δ.

Ch. 6: Lemma: Differentiability and continuity (6.22)

Differentiable functions are continuous

Ch. 6: Prop.: Continuous partial derivatives implies (6.23)

differentiable with derivative given by the Jacobian

Ch. 6: Prop.: Directional derivatives are determined by (6.24)

total derivatives.

Ch. 6: Corollary: If a function is differentiable and has partial derivatives (6.25)

the derivatives is the Jacobian.

Ch. 6: Def.: Operator norm (6.28)

et T : R^m → Rⁿ be a linear transformation. Define ∥T∥_op = sup_∥⃗x∥=1 {∥T ⃗x∥}

Ch. 6: Lemma: Estimate for operator norm (6.29)

Let T : R^m → Rⁿ be a linear transformation, which we will identify with its matrix with respect to the standard bases of R^m and Rⁿ. Then A = T^TT is a real symmetric (m × m)-matrix with real eigenvalues. Denote by λ_max the maximal eigenvalue of A. Then λ_max ≥ 0 and ∥T∥_op = √ λ_max.

Ch. 6: Prop.: Growth of differentiable functions (6.30)

is bounded; there exists δ > 0 such that ∥φ(⃗x) − φ(⃗p)∥ ≤ M∥⃗x − ⃗p∥ for all ⃗x ∈ X such that ∥⃗x − ⃗p∥ < δ where M > ∥(D φ)⃗_p∥_op.

Ch. 6: Prop.: Product Rule (6.32)

Let X ⊂ R^m be open, and let ⃗p ∈ X. If f, g : X → R are differentiable at ⃗p, then so is fg, with derivative (D fg)⃗_p = g(⃗p)(D f)⃗_p + f(⃗p)(D g)⃗_p.

Ch. 6: Prop.: Chain rule! (6.33)

Let X ⊂ R^m, Y ⊂ Rⁿ be open sets, and let ⃗p ∈ X. Let φ : X → Y be differentiable at ⃗p and let ψ : Y → R^s be differentiable at φ(⃗p). Then the composition (ψ ◦ φ) : X → R^s is differentiable at p with derivative (D(ψ ◦ φ))⃗_p = (D ψ)_φ(⃗p) ◦ (D ϕ)⃗_p. The derivative of a composition is the composition of the derivatives.

Ch. 7: Def. : Contraction (7.1)

Let X ⊂ R^m. We call a function φ : X → R^m a contraction if there exists 0 ≤ λ < 1 such that ∥φ(⃗u) − φ(⃗v)∥ ≤ λ∥⃗u − ⃗v∥ for all ⃗u, ⃗v ∈ X.

Ch. 7: (Weak) Banach Fixed Point Theorem (7.3)

Contractions have fixed points; “Let F ⊂ R^m be a closed subset, and let φ : F → F be a contraction. Then there is a unique point ⃗p ∈ F such that φ(⃗p) = ⃗p.”

Ch. 7: Def.: Continuously differentiable (7.7)

A function φ : X → Rⁿ defined on a open subset X ⊂ R^m is called continuously differentiable if it is differentiable with continuous first order partial derivatives throughout X. (by prop 6.23 any function with continuous first order partial derivatives throughout X is continuously differentiable on X)

Ch. 7: Inverse Function Theorem (7.9)

Let X ⊂ R^m be open, and let φ : X → R^m be a continuously differentiable function. Let ⃗p ∈ X be a point such that (D φ)⃗_p is invertible. Then there exists an open Y ⊂ R^m containing φ⃗(p) and a continuously differentiable function µ : Y → R^m such that µ(Y ) is open in R^m, ⃗p ∈ µ(Y ) and φ(µ(⃗y)) = ⃗y for every ⃗y ∈ Y .

Ch. 7: Implicit Function Theorem (7.13)

Let X ⊂ R^m be open, f₁, . . . , f_k : X → R be continuously differentiable functions on X, with k < m. Define S = {⃗x ∈ X | f_i(⃗x) = 0 for all 1 ≤ i ≤ k} and let ⃗p ∈ S. Suppose that the matrix J(⃗x) = (∂_jf_i(⃗x)) has rank at least k at ⃗p. Without loss of generality, we assume that the first k columns are independent, reordering the variables x₁, . . . , x_m if needed. Then there exists open V ⊂ X containing ⃗p, and continuously differentiable functions g₁, . . . , g_k : U → R defined on some open set U ⊂ R^m−k containing (p_k+1, p_k+2. . . . , p_m) such that S ∩ V = {(g₁(⃗x′ )), . . . , g_k(⃗x′ ), x_k+1, . . . , x_m) | ⃗x′ = (x_k+1, . . . x_m) ∈ U}

Ch. 8: Example: Failure of iterated integrals (8.5)

f : R² → R be f(x, y) := (4xy(x 2−y 2 )) / (x2+y2)³ if (x, y) ̸= (0, 0), 0 if (x, y) = (0, 0). Note that this is not continuous on [0, 1] × [0, 1], even though it is continuous on (0, 1] × (0, 1].

\int¹₀ ( \int¹₀ f(x,y) dy ) dx = ½ Noting that f(x, y) = −f(y, x), we must have that

\int¹₀ ( \int¹₀ f(x,y) dx ) dy = - \int¹₀ ( \int¹₀ f(y, x) dx ) dy = -1/2

Ch. 8: Fubini’s Theorem (8.7)

Let f : [a1, b1]×[a2, b2] → R be continuous. Then

\int^b2_a2 ( \int^b1_a1 f(x,y) dx ) dy = \int^b1_a1 ( \int^b2_a2 f(x,y) dy ) dx

Furthermore, if f is Riemann integrable, in the sense of limiting values of double Riemann sums, then its Riemann integral equal to the common value of the iterated integrals.