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MATH 351 Assignment
创建日期：2024-08-28 16:40:35
所属分类：经济Economics
标签： MATH 351
MATH 351 Assignment

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MATH 351


Assignment 

1. (On equivalence of norms) Let || · || and ||| · ||| be two norms defined on a R-vector space V . We let

(a) Verify that ≈ is an equivalence relation on the set N(V ) of norms on V .

(b) Consider the vector space R n . Show that for 1 < p < r < ∞ that

and deduce that these norms are all equivalent.

[Hint: for the hardest of these inequlities, if x ≠ 0, first divide ||x||rr by ||x||∞r, notice that t r ≤ t p if 0 ≤ t ≤ 1.]

(c) Show that for x in R n we have ||x||∞ = limp→∞ ||x||p.

(d) Show that for 1 ≤ p < r < ∞ we have a proper containment relations

[The integral test for series convergence is your friend.]

(e) Show that for 1 ≤ p < r ≤ ∞, || · ||p 6≈ ||· ||r on ` 1. (Recall, from above, that ` 1 ? ` p for each 1 ≤ p ≤ ∞.)

2. (On equivalence of metrics)

Let X be a non-empty set and M(X) ? [0,∞) X×X denote the set of all metrics on X. For d, ρ in M(X) let

(a) Verify that ≈ and ～ are equivalence relations on M(X).

(b) Show that d ≈ ρ in M(X) implies that d ～ ρ

(c) Show that d ～ ρ in M(X) ? (X, d) and (X, ρ) admit the same open sets.

(d) Let d ∈ M(X) and f : [0,∞) → [0,∞) satisfy that

f(0) = 0 and f is strictly increasing, subadditive and continuous. (?)

Show that df : X × X → [0, ∞), df (x, y) = f(d(x, y)) defines a metric with df ～ d. Furthermore, show that f(s) = s+1/s satisfies (?).

(e) Does d ～ ρ in M(X) imply that d ≈ ρ? Prove this, or supply a counterexample.

3. We say that a metric space (X, d) is separable if there is a countable set Z = {zk} ∞k=1 ? X with closure Z = X.

(a) Show that if (X, d) is separable, then it cardinality satisfies |X| ≤ c.

[Hint: this has aspects similar to our construction of R from Q.]

(b) Show that for 1 ≤ p < ∞ that ` p is separable.

(c) Let C(R) = {f ∈ R R : f is continuous}. Show that |C(R)| = c.

[Hint: a continuous function is determined by its behaviour on Q.]

(d) Show that ` ∞ is not separable.

[Hint: first find a subset X of elements which is uncountable and for which k χ ? χ 0 k ∞ ≥ 1 for χ = χ 0 in X.]

(e) Show that |` ∞| = c.

4. In R with usual metric d(x, y) = |x ? y| let

(a) Compute the interior A? , boundary ?A, derived set A0 , and closure A, in R.

(b) Let now B = {? k/1 : k ∈ N}. Compute the interior, boundary, derived set, and closure but relativized in A. [You might wish to write, B?A, ?AB, B'A, BA for these relativized sets.]

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