Advance Calculus 1 Test 2 Mathematics Exercise Help

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I need help with this sample test for my advance calculus class. The test is attached bellow.

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Advance Calculus I Test 2 Name:________________________________ Time allowed: 50 minutes Show your working. All questions carry equal number of points. 1. a. Let V be a vector space. Define a norm on V. b. Prove that if x x 1 , x 2, . . . , x p R p , then x 1 sup |x 1 |, . . . , |x p | is a norm on R p . 2. a. Define a set G to be open in R p . b. Prove that the intersection of two open sets is open. c. Prove that the union of any collection of open sets is open. 3. Prove the following nearest point theorm: Let F be a non-empty set in R p , and let x be a point y x for outside F. Then there exists at least one point y belonging to F such that z x every z F. 4. Prove that a sequence x n in R p with x n x 1n , x 2n , . . . , x pn converges to y y 1 , y 2 , . . . , y p if and only if the corresponding sequences of real numbers x 1n , x 2n , . . . , x pn converges to y 1 , y 2 , . . . , y p respectively.
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Running head: Advance calculus 1 test 2

1

Advance calculus 1 test 2
Name of student
Name of professor
Name of course
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Advance calculus 1 test 2
Question one
Part a
Let V be a vector space. Define a norm on V.
A norm on V is defined as a real-valued function ||. ||: V→R which satisfies;
Positivity condition given by ||v|| ≥ 0 for all ∈ V and ||v|| = 0 if and only if v = 0.
Scaling condition given by||αv|| = |α| ||v|| for all scalars α and all vectors v ∈ V.
Triangular inequality condition given by ||u +...


Anonymous
This is great! Exactly what I wanted.

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