Reference no: EM13966703
1. Let (S, d) be any noncompact metric space. Show that there exist bounded continuous functions fn on S such that fn (x ) ↓ 0 for all x ∈ S but fn do not converge to 0 uniformly. Hint: S is either not complete or not totally bounded.
2. Show that a metric space (S, d) is complete for every metric e metrizing its topology if and only if it is compact. Hint: Apply Theorem 2.3.1. Suppose d(xm, xn ) ≥ ε > 0 for all m /= n integers ≥ 1. For any integers j, k ≥ 1 let e jk (x, y) := d(x, x j ) +| j -1 - k-1|ε + d(y, xk ).
Let e(x, y) := min(d(x, y), inf j,ke jk (x, y)). To show that for any j, k, r , and s, and any x, y, z ∈ S, e js (x, z) ≤ e jk (x, y) + ers (y, z), consider the cases k = r and k /= r .
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Subsequences of the an may converge
: Let F be a closed set in R whose complement is the union of disjoint open intervals (an, bn ) from part (a). Let f be a continuous real-valued function on F . Extend f to be linear on [an, bn ], or if an or bn is infinite, extend f to be constant ..
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Debate the issues professionally and provide examples
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Calculate the complex power s consumed by this load
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Problem regarding the bounded continuous functions
: Let (S, d) be any noncompact metric space. Show that there exist bounded continuous functions fn on S such that fn (x ) ↓ 0 for all x ∈ S but fn do not converge to 0 uniformly. Hint: S is either not complete or not totally bounded.
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In what ways does willy loman recognize his own shortcomings
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How they distribute their products.
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Curve on a finite interval
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Irrational numbers with usual topology
: 1. Show that the set R\Q of irrational numbers, with usual topology (relative topology from R), is topologically complete. 2. Define a complete metric for R\{0, 1} with usual (relative) topology.
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