MAT 480 Topology Problem Set 1 Due 2/1/2019 Spring 2019 1. Let a (x1, z2, 23) and y - (1, 2, va) be elements of R3. Define d(x,y) = maxlx1-vil, la 2-U21, las-val). Prove that d defines a metric on R3 2. Let u(u1, u2, ..., un) and - (v, V2, ..., Vn) be vectors in Rn. Prove the Cauchy-Schwarz Inequality lin 히 lizil . Ilull in six easy steps: (a) Explain why lu.卵く间2·间2 implies lu. 히 (b) Explain why, for any real z, 阅·间. (c) Rewrite the right-hand side of the inequality above as a quadratic polynomial az2 + baz + c. What are the values of the coefficients a, b, and c? (d) Let A-b2 4ac. Explain why A s 0. Hint: (b) implies 0 az2 ba + c; what do you know about the graph of such a quadratic polynomial'? (e) Use (d) to show lž.2I2. l12 (f) Conclude that the Cauchy-Schwarz Inequality is true. 3. Given a metric space (X, d), a point a E X, and some e> 0, we can define the "open" e-ball centered at as Similarly, the "closed" e-ball is defined as B(r, e)(v E 지 d(x, y) e). (a) Let de be the Euclidean metric on R2. Sketch a picture of B(0,0), 1) in (R2, dB) (b) Let dr be the Taxi-cab metric on IR2. Sketch a picture of B((0,0), 1) in (R2, dr). (c) True or false, with explanation: given B((0,0),1) in (R2, dE) and any point V E B((0,0), 1), there is always some 6> 0 so that (d) Do we have the same answer if we replace B(0, 0), 1) with 瓦(0,0), 1) in (c)? Explain.

Question

I am taking a Topology course and was wondering if you could help me with Problem 2 and 3 of the worksheet that is attached.

MAT 480
Topology
Problem Set 1
Due 2/1/2019
Spring 2019
1. Let a (x1, z2, 23) and y - (1, 2, va) be elements of R3. Define
d(x,y) = maxlx1-vil, la 2-U21, las-val).
Prove that d defines a metric on R3
2. Let u(u1, u2, ..., un) and - (v, V2, ..., Vn) be vectors in Rn. Prove the Cauchy-Schwarz
Inequality lin 히 lizil . Ilull in six easy steps:
(a) Explain why lu.卵く间2·间2 implies lu. 히
(b) Explain why, for any real z,
阅·间.
(c) Rewrite the right-hand side of the inequality above as a quadratic polynomial
az2 + baz + c. What are the values of the coefficients a, b, and c?
(d) Let A-b2 4ac. Explain why A s 0. Hint: (b) implies 0 az2 ba + c; what do you
know about the graph of such a quadratic polynomial'?
(e) Use (d) to show lž.2I2. l12
(f) Conclude that the Cauchy-Schwarz Inequality is true.
3. Given a metric space (X, d), a point a E X, and some e> 0, we can define the "open" e-ball
centered at as
Similarly, the "closed" e-ball is defined as
B(r, e)(v E 지 d(x, y)
e).
(a) Let de be the Euclidean metric on R2. Sketch a picture of B(0,0), 1) in (R2, dB)
(b) Let dr be the Taxi-cab metric on IR2. Sketch a picture of B((0,0), 1) in (R2, dr).
(c) True or false, with explanation: given B((0,0),1) in (R2, dE) and any point
V E B((0,0), 1), there is always some 6> 0 so that
(d) Do we have the same answer if we replace B(0, 0), 1) with
瓦(0,0), 1) in (c)? Explain.
Expand
Transcribed Image Text

MAT 480 Topology Problem Set 1 Due 2/1/2019 Spring 2019 1. Let a (x1, z2, 23) and y - (1, 2, va) be elements of R3. Define d(x,y) = maxlx1-vil, la 2-U21, las-val). Prove that d defines a metric on R3 2. Let u(u1, u2, ..., un) and - (v, V2, ..., Vn) be vectors in Rn. Prove the Cauchy-Schwarz Inequality lin 히 lizil . Ilull in six easy steps: (a) Explain why lu.卵く间2·间2 implies lu. 히 (b) Explain why, for any real z, 阅·间. (c) Rewrite the right-hand side of the inequality above as a quadratic polynomial az2 + baz + c. What are the values of the coefficients a, b, and c? (d) Let A-b2 4ac. Explain why A s 0. Hint: (b) implies 0 az2 ba + c; what do you know about the graph of such a quadratic polynomial'? (e) Use (d) to show lž.2I2. l12 (f) Conclude that the Cauchy-Schwarz Inequality is true. 3. Given a metric space (X, d), a point a E X, and some e> 0, we can define the "open" e-ball centered at as Similarly, the "closed" e-ball is defined as B(r, e)(v E 지 d(x, y) e). (a) Let de be the Euclidean metric on R2. Sketch a picture of B(0,0), 1) in (R2, dB) (b) Let dr be the Taxi-cab metric on IR2. Sketch a picture of B((0,0), 1) in (R2, dr). (c) True or false, with explanation: given B((0,0),1) in (R2, dE) and any point V E B((0,0), 1), there is always some 6> 0 so that (d) Do we have the same answer if we replace B(0, 0), 1) with 瓦(0,0), 1) in (c)? Explain.

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