Honors Exam 2011 Real Analysis

Part I: Real Analysis

1. A topological space is called separable if it contains a countable dense subset.

(a) Show that euclidean space Rⁿ is separable.

(b) Show that every compact metric space is separable.

(c) Let l_∞ be the space of bounded sequences a = {a_j}^∞_j=1. We make l_∞ into a metric space by declaring

d(a, b) = sup_j|a_j - b_j|

Show that l_∞ is not separable.

2. Let {f_n} be a uniformly bounded sequence of continuous functions on [a, b]. Let

F_n(x) = _a∫^x f_n(t)dt

(a) Show that there is a subsequence of {F_n} that converges uniformly on [a, b].

(b) Each F_n is differentiable. Show by an example that the uniform limit in part (a) need not be differentiable.

3. Let {a_n}^∞_n=1 be a positive decreasing sequence a_n ≥ a_n+1 ≥ 0.

(a) Show that _n=1∑^∞ a_n converges if and only if _k=0∑^∞2^ka_2^k converges.

(b) Use the result in part (a) to show that the harmonic series _n=2∑^∞(1/n) diverges.

(c) Use the result in part (a) to show that the series _n=2∑^∞(1/n(log n)^p) converges for p > 1.

4. Let f be a continuous function on the closed interval [a, b].

(a) Show that

lim_p→∞( _a∫^b|f(x)|^pdx)^1/p = max_x∈[a,b]|f(x)|

(b) Give an example of a continuous function f on (a, b) where the improper integrals

_a∫^b|f(x)|^pdx

exist (i.e. are finite) for all 1 ≤ p < ∞, but

limp→∞ (_a∫^b |f(x)|^pdx)^1/p = ∞

Part II: Analysis on Manifolds

1. On R³ - {0}, consider the following 2-form

ω = (x dy ∧ dz + y dz ∧ dx + z dx ∧ dy/(x² + y² + z²)^3/2)

(a) Show that ω is closed.

(b) By explicit computation show that

∫_S^2ω

does not vanish. Here, S² ⊂ R³ is the unit sphere.

(c) Is ω exact?

2. The space M(n) of n × n matrices is a manifold naturally identified with Rn². The subspace S(n) of symmetric matrices is a manifold naturally identified with R^n(n+1)/2. The space of orthogonal matrices is

O(n) = {A ∈ M(n) : AA^T = I}

where T denotes transpose and I is the identity matrix.

(a) Show that O(n) is manifold by considering the map F: M(n) → S(n) : A |→ AA^T

(Hint: it might be easier to first consider the derivative of F at the identity, and then use matrix multiplication).

(b) What is the dimension of O(n)?

(c) Describe the tangent space of O(n) at the identity matrix as a subspace of M(n).

3. Let A ⊂ R^m, B ⊂ Rⁿ be rectangles, Q = A × B, and f: Q → R a bounded function.

(a) State necessary and sufficient conditions for the existence of the Riemann integral

∫_Qf(x, y) dxdy

(b) Suppose ∫_Q f(x, y) dxdy exists. Show that there is a set E ⊂ A of measure zero such that for all x ∈ A - E, the Riemann integral

∫_{x}×Bf(x, y) dy

exists.

4. Let M be a compact, oriented manifold without boundary of dimension n, and let S ⊂ M be an oriented submanifold of dimension k. A Poincar´e dual of S is a closed (n - k)-form η_S with the property that for any closed k-form ω on M,

∫_Mω ∧ η_S = ∫_Sω

(a) Show that if η_S is a Poincar´e dual of S, so is η_S + dα for any (n - k - 1)-form α.

(b) Find a Poincar´e dual of a point.

(c) Let S ⊂ M be an embedded circle S = S¹ in an oriented 2-dimensional manifold M. Find a Poincar´e dual to S (Hint: use the fact that there is a neighborhood of S in M of the form S¹ × (-1, 1)).

5. Let M be a differentiable manifold. If X is a smooth tangent vector field on M, then X gives a map C^∞(M) → C^∞(M) on smooth functions by f |→ X(f) = df(X). Explicitly, in local coordinates x₁, . . . , x_n,

X = _i=1∑ⁿXⁱ ∂/∂x_i,                                X(f) = _i=1∑ⁿXⁱ ∂f/∂xⁱ

(a) Show that if X₁ and X₂ are vector fields such that X₁(f) = X₂(f) for all f ∈ C^∞(M), then X₁ = X₂.

(b) If X, Y are vector fields on M, define the Lie bracket [X, Y] to be the vector field determined by the condition that

[X, Y](f) = X(Y (f)) - Y (X(f))

for all f ∈ C^∞(M). Compute [X, Y] in local coordinates.

(c) Show that if N ⊂ M is a smooth submanifold of M and X, Y are two tangent vector fields to N, then [X, Y] is also tangent to N.