Lecture Notes Week 1 - Real Analysis PDF

# Some Notation and Definitions on R^n - For E = {1, 2,..., n} - R = R x R x ... x R - n-copies - x = (x1, x2,..., xn) ∈ R^n - O = (0, 0,..., 0) ∈ R^n - x ∈ R^n, λ ∈ R - x + y = (x1 + y1, x2 + y2, ..., xn + yn) ∈ R^n - dx = (dx1, dx2, ..., dxn) ∈ R^n - For x, y ∈ R^n, we define the inner product...

# Some Notation and Definitions on R^n - For E = {1, 2,..., n} - R = R x R x ... x R - n-copies - x = (x1, x2,..., xn) ∈ R^n - O = (0, 0,..., 0) ∈ R^n - x ∈ R^n, λ ∈ R - x + y = (x1 + y1, x2 + y2, ..., xn + yn) ∈ R^n - dx = (dx1, dx2, ..., dxn) ∈ R^n - For x, y ∈ R^n, we define the inner product on R^n by: <x, y> = sum_{i=1}^{n} x_iy_i = x1y1 + x2y2 + ... + xnyn - Then <•, •> satisfies: - <x, x> = x1^2 + x2^2 + ... + xn^2 >= 0 - <x, x> = 0 iff x = 0 - for λ, β ∈ R, x, y, z ∈ R^n - <λx + βy, z> = λ<x, z> + β<y, z> - <x, λy + βz > = λ<x, y> + β<x, z> - (<•, •>) is a bilinear map and is called the "inner product" - For x ∈ R^n, write ||x|| = sqrt(x1^2 + x2^2 + .... + xn^2). - Then ||x|| = sqrt(<x, x>) - For x, y ∈ R^n, then | <x, y>| ≤ ||x|| ||y|| - This is called the Cauchy-Schwarz inequality. - If x=0, y!=0, then |<x, y>|=0 and ||y||!=0 - We need to prove only when ||x||=||y||=1 - For t∈R, <x-ty, x-ty> = ||x-ty||^2 >= 0 - Let p(t) = <x-ty, x-ty> for t∈R - p(t) = <x, x>-2t<x, y>+t^2<y, y> - p(t) = t^2-2t<x, y>+1>=0 - Take t0 = <x, y>, p(t0)=0 - t0^2-2t0<x, y>+1 = 0 - So t0^2 - 2t0<x, y> + <x, y>^2 >= 0 - (t0-<x, y>)^2 >=0 this shows that - |<x, y>| ≤ ||x|| ||y|| - Also, |<x, y>| = ||x|| ||y|| iff x, y ∈ R^n and x = λy - (Linearly dependent sets explanation) # Maximum & Minimum - For x, y∈R^n, - ||x + y||^2 = <x + y, x + y> - ||x + y||^2 = ||x||^2 + ||y||^2 + 2<x, y> - ||x + y||^2 ≤ ||x||^2 + ||y||^2 + 2||x|| ||y|| - ||x + y||^2 ≤ ( ||x|| + ||y|| )^2 - ||x + y|| ≤ ||x|| + ||y|| - (Triangle inequality) - **Infimum & Supremum:** - R = (-∞, ∞), A⊆R - Given c, b ∈ R where a≤ c ≤ b and b ≤ g, then one can think of max & min of the set A. - A = {1, 2,..., 100} - maxA = 100 - minA=1 - However, for A = {..., -2 ,-1, 0}? - maxA = ∞ - minA = -∞ - **Infimum & Supremum:** - a1, a2 ... a_n ∈ A - **Definition:** - α ∈ R is called infimum (or glb) of the set A ⊆ R, if - α ≤ a for all a ∈ A - For all ε > 0, a ∈ A such that α + ε > a - (α is not a lower bound) - **Definition:** - β ∈ R is called supremum of the set A, if - α ≥ a for all a ∈ A - for all ε > 0, β - ε < a for some a ∈ A - (β is not an upper bound of A) # Bolzano-Weierstrass Theorem - Every bounded sequence (a_n) ∈ R has a convergent subsequence. - f : N → R - {f(1), f(2), f(3), ... } - (a_n) convergent sequence - (a_n) → a - a_1, a_2, a_3,... a_n, a_(n+1) - (..., a_t, a_(t+1), ...) - Let (a_t, a_(t+1),...) ⊂ N for t ∈ N - a_t < a_n < a_(t+1) for n > t, t ∈ N, - So (a_n) converges to a - E.g. a_n = 1/n, expect what would be the limit? - **Necessary facts about convergent sequences** - For ε>0, ∃N∈N such that - |a_n - a |< ε, ∀ n > N - We can say a_n ∈ (a-ε, a+ε) for n >N - We can say that a_t =(a-1, aH) for t ∈ N. - (a is a sup(a-1, aH) = a+1, aT∈N - - **Explanation:** - t0 = sup(a-1, aH) = a + 1 for aT∈N - If a_n is convergent then a_n is bounded. - E.g. a_n = {1, 2, 3,...} - {a_n ∈ R : ε>0} a family of lower bound - {a_n ∈R : ε>0} a family of upper bound - **Proof:** - If a_n is convergent, then we can say a_n <= (sup a_n <= a-ε - where ε is very small. - ε = 1/2 - |a_n - a| < ε - **Proof:** - ε =1/2 - ε = 1/2, d_n = a_n – a - a_n = a + d_n - sup d_n = a - a =0 - **Proof:** - 1/2 = ε - (a_n) converges - ε < d_n < ε - d_n = a_n - a - (a_n) converges, -ε < d_n < ε (since d_n = a_n - a) - For |a_n - a| < & for n > N - a = lim(a_n) - **Bolzano-Weierstrass theorem:** - For (a_n) ∈ R - Every bounded set has a convergent subsequence. - **Proof:** - (a_n) is bounded - sup(a_n) = α, sup(a_n) = β - a_n ∈[α, β] - a_n < α for n >0 - (a_n) convergent, (a_n) → α. - a = lim(a_n) # Bolzano-Weierstrass Theorem for R^n - lim(a_n, a_n2, a_n3...) = (x, y, z...) =x - a_n1, a_n2,... a_n - lim(a_nk) = x for k = 1, 2, 3... - **Bolzano-Weierstrass Theorem for R^n** - Let {x_n} = {(x_n, y_n)}^∞_{n=1} - ||x_n|| = sqrt(x_n^2+y_n^2) <= M, ∀n∈Z - When considering x_n^2 + y_n^2, there are a finitely bounded number of sets - ∀n∈Z, x_n^2 + y_n^2 <= M^2 - There are a finitely bounded number of sets that are closed because x_n^2 + y_n^2 ≤ M^2 - (x_nk, y_nk) ⊂ (x_n, y_n) - (x_nk, y_nk) → (x, y) - **Important Note:** - |a_n - a| ≤ ε, for n >N - Every convergent sequence has a convergent subsequence. - If {x_k} is a bounded sequence in R^n - {x_k} is a bounded sequence in R^n - Then we can say: - x = (2x^k, x^k, ..., x^k) - If {x_k} is a bounded sequence in R^n. - Then it has a subsequence (x_k) -> (x, y, z... ) ∈ R^n - **Ex:** - Let a_n ∈ R, {a_n} ↑ and bounded above, then (a_n) is convergent to sup(a_n). - Let sup (a_n) = B - For ε > 0, since a_n is bounded above - For ε>0, take a_n such that B-ε < a_n < a_(n+1) < B... - B-ε < a_n < B - B-ε < a_n < B for n>N - (a_n) is bounded below, then a_n is convergent. - **Closed sets & Open sets for R:** - **Definition**: A set F ⊆ R is said to be closed if for any sequence a_n ∈ F such that a_n -> a ∈ R ⇒ a ∈ F. - (The limit of a_n is a continuous function such that the limit of a_n is contained in F) - A Set O ⊆ R is said to be open in R if O is not closed - **Definition:** - O ⊆ R is open if for any a ∈ O, Ξ ε > 0 such that ( a- ε, a+ ε) ⊆ O - (if not for some ε > 0, then (a-ε, a + ε) ⊈ O) - **Proof:** - For any x ∈ O, ε = Y_m, ∃ n ∈ N such that (x-maxY_m, x+maxY_m) ∩ O_c. - This means x is an interior point so x_n -> x as n -> ∞, such that x is a closed set - Thus, x is a convergent sequence on O. - x ∈ O is not closed - Therefore O is open. - **Note**: - F is closed if & only if F^c is open. - **Definition:** - A set O ⊆ R is open if for any a∈O, ∃ ε>0 such that (a- ε, a+ε) ⊆ O. - **Ex.** - A = {1, 1/ 2, 1/3, ... }, A is not open & not closed. - All I = (a, b) are open. - **Proofs:** - If I ∈ R, then I^c is closed, I^c is closed. - If I ∈ R, then I is open, I^c is open - If I ∈ R, then U_i is closed, then U_i is closed - If I ∈ R, then ∩U_i is closed, then ∩U_i is closed (this is not true - ∩ (a_i, b_i) = {∅} : Ex. (-∞, c_i) ∪ (c_i, ∞) - (The countable intersection of open sets need not be open) - Let I_n = (-1/n, 1/n). I_n is closed - ∩ I_n = {0} - Let O_n = (1/n, ∞). O_n is not closed. - ∩ O_n = {} for n ∈ R ## Open set in R^n - Let I = (a, b) ⊆ R^n, I_ij∈Z, i≠j, if i≠j - Then, I is open - **Result**: Any open set O⊆R^n is countable union of open disjoint intervals. - **Proof:** - x∈O ⇒ (x-ε, x+ε) ⊆ O - Choose largest open interval containing x. - a_x = inf{x ∈R : (a, x) ⊆ O} - b_x = sup{b ∈ R : (x, b) ⊆ O} - X_e = (a_x, b_x) = I_x (⊆ O) - If y∈(a_y, b_y) = I_y (⊆O) - If x≠y, assume a_x < a_y, then by definition of a_x, (a_x, a_y) ⊂ O. Then by definition of b_x, (a_x, b_y) ⊂ O and X_e ∩ X_y = {∅} - Then by definition of a_x and b_y, (a_x, b_x) ∩ (a_y, b_y) = {∅}. - Therefore, x ∈O ⇒ x ∈ I_x = (a_x, b_x ) ⊆ O - Then, ∪I_x ⊆ O & ∪I_x ⊆ O. - Choose x_i∈I_x ∩ Q : (Q - rationals) - Then x → x_i as i → ∞. - Since I_n ∩ I_m = {∅} - O = ∪{I_x} - where x ∈ Q