MATH 381 Midterm: MATH 381 McGill Examw99

• School: McGill University
• Department: Mathematics and Statistics
• Course: MATH 381
• Professor: All

Document Summary

Mathematics 189-354a: carefully state the following theorems. (i) (5 marks) the baire category theorem. (ii) (5 marks) the stone weierstrass theorem. (iii) (5 marks) the implicit function theorem. (iv) (5 marks) the picard existence theorem. 2. (i) (5 marks) write down the de nition of the term metric space. (ii) (5 marks) write down the de nition of the term open subset . (iii) (5 marks) de ne the geodesic distance on the unit sphere. {a + b|a a, b b} is again closed. (iv) (5 marks) show that the set {a + b|a a, b b} need not be closed if a and. B are assumed closed, but not necessarily bounded. Suppose that x1 and x2 are distinct components of x. Show that there is a splitting of x for which x1 and x2 lie on di erent sides.