Real Analysis II.

Instructor: Dmitry Ryabogin

Assignment X.

1. Problem 1.

In class we considered the determinant D_{m} of the matrix, (d_{ij})_{1≤i,j≤m}, d_{ij} = (a_{i} +b_{i})^{−1}, and showed that

D_{m} = P_{m}/C_{m}, C_{m} = Π_{i,j=1}^{m}(a_{i} + b_{j}),

where P_{m} is a polynomial of a_{i} and b_{j}.

a) Prove that deg(P_{m}) ≤ m^{2} − m.

b) Prove that deg(A_{m} B_{m}) = m^{2} − m,

A_{m} B_{m} = Π_{1≤i<j≤m}(a_{i} − a_{j}) Π_{1≤i<j≤m}(b_{i} − b_{j}).

c) Prove that

lim_{a_{m}→∞} lim_{b_{m}→∞} \frac{a_{m} Π_{1≤i<j≤m}(a_{i} − a_{j})(b_{i} − b_{j})}{Π_{i,j=1}^{m}(a_{i} + b_{j})} =

\frac{Π_{1≤i<j≤m−1}(a_{i} − a_{j})(b_{i} − b_{j})}{Π_{i,j=1}^{m−1}(a_{i} + b_{j})}.

2. Problem 2. A point x_{0} ∈ C ⊆ H is called an internal point of C if for every x ≠ x_{0} there exists ϵ > 0 such that for every 0 ≤ λ < ϵ we have (1 − λ)x_{0} + λx ∈ C.

a) Prove that every interior point is also an internal point.

b) Let x be an internal point of convex C, and let y ∈ C. Then for all 0 ≤ λ ≤ 1, λx + (1 − λ)y is an internal point of C.

c) Find a set C ⊂ R^{2} such that the origin is an internal point of C but is not its interior point.

d) Let {y_{α}}_{α∈A} be an algebraic basis of H, and let ∥y_{α}∥ = 1 ∀α. Pick a sequence (x_{n})_{n=1}^{∞} of (countably many) vectors from the basis, and define C to be a set of vectors in H such that in the decomposition y = ∑_{k=1}^{m} c_{α_{k}}y_{α_{k}} , m = m(y), the coefficient near x_{n} is less than 1/n in absolute value. Prove that C is convex, that the origin is an internal point of C, but is not its interior point.

3. Problem 3.

Let g_{1}, g_{2}, ..., g_{n} be vectors in R^{n}, g_{k} = (g_{k}(1), ..., g_{k}(n). It is known that

D(g_{1}, ..., g_{n}) = |det{(d_{ij})_{1≤i,j≤n}}|, d_{ij} = g_{j}(i),

is a volume of a parallelepiped Π_{n}(g_{1}, ..., g_{n}) generated by vectors g_{1}, ..., g_{n}.

a) Prove that G(g_{1}, ..., g_{n}) = (D(g_{1}, ..., g_{n}))^{2}, and conclude that G(g_{1}, ..., g_{n}) = 0 iff g_{1}, ..., g_{n} are linearly dependent.

b) Consider R^{n} as a sub-space of R^{n+1}, i.e.,

R^{n} = {x ∈ R^{n+1} : x = (x(1), x(2), ..., x(n), 0)}.

Let g_{1}, ..., g_{n} ∈ R^{n}, x ∈ R^{n+1}, and let M = span{g_{1}, ..., g_{n}}, d := d(x, M) = inf_{y∈M} ∥x − y∥. Prove that

d^{2} G(g_{1}, ..., g_{n}) = G(x, g_{1}, ..., g_{n}).

Hint:

vol_{n+1}(Π_{n+1}(x, g_{1}, ..., g_{n})) = vol_{n}(Π_{n}(g_{1}, ..., g_{n})) height.

4. Problem 4. Let α > 0, f_{α}(x) = e^{αx}, and let H = L^{2}(−∞, 0). Compute

dist(f_{α_{0}}, span{f_{α_{1}}, ..., f_{α_{m}}}),

provided α_{j}, j = 0, ..., m are pairwise different.