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Exercise 15. Are the following functions norms on the vector spaces they are defined? Prove your...
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in...
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in class is a valid norm on R2 (b) Prove that in R2, any open ball in 12 ("Euclidean metric") can be enclosed in an open ball in the loo norm ("sup" norm). (c). Say I have a collection of functions f:I R. Say I (1,2). Consider the convergence of a sequence of functions fn (z) → f(x) in 12-Show that the convergence amounts to...
Please answer C 3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which...
Please answer C
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
Please answer D 3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which...
Please answer D
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
EXERCISE 6.3.8 For each of the following spaces, which has a deformation retract of (i) a point, ...
EXERCISE 6.3.8 For each of the following spaces, which has a deformation retract of (i) a point, (ii) a circle, (iii) a figure eight, or (iv) none of these? a. R3 minus the nonnegative x, y, and z axes b. R2 minus the positive x axis c. S U(r, 0)-1 <x <1], where Sl is the unit circle in the plane d. IR3 e. S2 minus two points f. R2 minus three points g. S2 minus three points h. T...
- Let V be the vector space of continuous functions defined f : [0,1] → R...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
6. Are the following vector spaces over R. (c) RQ (irrational numbers) (а) Q (d) V...
6. Are the following vector spaces over R. (c) RQ (irrational numbers) (а) Q (d) V {(, y) E R2: y 2 (Ъ) М, (R) (e) R2 however, we define (v, v2) (wi, w2) (U1w1, v2W2) (f) C(I), the continuous functions on the interval I.
6. Are the following vector spaces over R. (c) RQ (irrational numbers) (а) Q (d) V {(, y) E R2: y 2 (Ъ) М, (R) (e) R2 however, we define (v, v2) (wi, w2) (U1w1,...
3. This example hopes to illustrate why the vector spaces the linear transformation are defined o...
3. This example hopes to illustrate why the vector spaces the linear transformation are defined on are critical to the question of invertibility. Let L : → p, be defined by L(p)(t+1)p(t)-plt). (a) Given a basis of your choice, find a matrix representation of I with respect to your chosen basis (b) Show L: P+P is not invertible (e) Let V-span+21-4,+2t-8). It can be shown that L VV. Given an ordered basis for V of your choice, find a matrix...
15. Advanced problem: Let's say that a vector space X“splits” the spaces U and G if...
15. Advanced problem: Let's say that a vector space X“splits” the spaces U and G if either Uç X & W or W ÇX V. a. Is there a vector space C that splits A = R^3 and B = {the x-axis in R^3} ? If there is, find it (no need to prove your claim) and if not, explain why it cannot exist. b. Suppose that U & W are a finite-dimensional. On what condition does there exist a...
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 fo...
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,...,...
real analysis
hint
9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,..., defined as follows: X E if and only if II x11 if and only if lxsup lloo. for which ξί (a) If y = {nJ E 11 and Ax = Σ ζίηǐ for every x ε co, then Λ is a bounded linear functional on (More precisely, these two spaces are not equal; the preceding statement exhibits...
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in class is a valid norm on R2 (b) Prove that in R2, any open ball in 12 ("Euclidean metric") can be enclosed in an open ball in the loo norm ("sup" norm). (c). Say I have a collection of functions f:I R. Say I (1,2). Consider the convergence of a sequence of functions fn (z) → f(x) in 12-Show that the convergence amounts to...
Please answer C
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
Please answer D
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
EXERCISE 6.3.8 For each of the following spaces, which has a deformation retract of (i) a point, (ii) a circle, (iii) a figure eight, or (iv) none of these? a. R3 minus the nonnegative x, y, and z axes b. R2 minus the positive x axis c. S U(r, 0)-1 <x <1], where Sl is the unit circle in the plane d. IR3 e. S2 minus two points f. R2 minus three points g. S2 minus three points h. T...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
6. Are the following vector spaces over R. (c) RQ (irrational numbers) (а) Q (d) V {(, y) E R2: y 2 (Ъ) М, (R) (e) R2 however, we define (v, v2) (wi, w2) (U1w1, v2W2) (f) C(I), the continuous functions on the interval I.
6. Are the following vector spaces over R. (c) RQ (irrational numbers) (а) Q (d) V {(, y) E R2: y 2 (Ъ) М, (R) (e) R2 however, we define (v, v2) (wi, w2) (U1w1,...
3. This example hopes to illustrate why the vector spaces the linear transformation are defined on are critical to the question of invertibility. Let L : → p, be defined by L(p)(t+1)p(t)-plt). (a) Given a basis of your choice, find a matrix representation of I with respect to your chosen basis (b) Show L: P+P is not invertible (e) Let V-span+21-4,+2t-8). It can be shown that L VV. Given an ordered basis for V of your choice, find a matrix...
15. Advanced problem: Let's say that a vector space X“splits” the spaces U and G if either Uç X & W or W ÇX V. a. Is there a vector space C that splits A = R^3 and B = {the x-axis in R^3} ? If there is, find it (no need to prove your claim) and if not, explain why it cannot exist. b. Suppose that U & W are a finite-dimensional. On what condition does there exist a...
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
real analysis
hint
9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,..., defined as follows: X E if and only if II x11 if and only if lxsup lloo. for which ξί (a) If y = {nJ E 11 and Ax = Σ ζίηǐ for every x ε co, then Λ is a bounded linear functional on (More precisely, these two spaces are not equal; the preceding statement exhibits...
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