(3 points.) Prove that the following properties hold in any vector space: (1) (-с)и %— —...
Пло и шелу и е ишити јошли пет е и есту е силу - с 2 теталу. Problem 28. Let (X.) be any metric space. Prove that a Cauchy sequence converges if and only if it has a convergent subsequence.
(Hz CH3 I Cexcers) 2) лого нао 1)CH₂ I (excess) - си, 2) 2 - 3 ICH₃ I. mcPPA си с из сен и 2. А
11. Prove that the identity vector in any vector space is unique. (Hint: use contradiction) 12. Find bases for Nul A and Col A. (8pts) 1 5 3 1 - 1 2 22 5 0 - 8 - 24 -48 3 - 2
1. [4-+6+6-16 points Let /°0 denote the vector space of bounded sequences of real numbers, with addition and scalar multiplication defined componentwise. Define a norm Il on by Il xl = suplx! < oo where x = (x1,x2, 23, . .. ) iEN (a) Prove that is complete with respect to the norm | . (b) Consider the following subspaces of 1o i) c-the space of convergent sequences; (i) co-the space of sequences converging to 0; (iii) coo- the space...
6. (i) Prove that if V is a vector space over a field F and E is a subfield of F then V is a vector space over E with the scalar multiplication on V restricted to scalars from E. (ii) Denote by N, the set of all positive integers, i.e., N= {1, 2, 3, ...}. Prove that span of vectors N in the vector space S over the field R from problem 4, which we denote by spanr N,...
linear algebra
5. Let V be a vector space and let x,yeV. Show that you can prove property C (commutative).i.e., x+y- yx from the other properties of vector spaces by computing (I+)x) two different ways using DSA and DVA
5. Let V be a vector space and let x,yeV. Show that you can prove property C (commutative).i.e., x+y- yx from the other properties of vector spaces by computing (I+)x) two different ways using DSA and DVA
Topology
C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...
Exercise 15. Are the following functions norms on the vector spaces they are defined? Prove your answer. (i 21 - 3|r2 for x - (71 12)Т € R?. (1x2)f(x)|d for f(x) e C[0, 1] (i) _ (iii) pl dо + 2la| + 3/az| for p(z) — аz2? + ајя + ao є Pз.
Exercise 15. Are the following functions norms on the vector spaces they are defined? Prove your answer. (i 21 - 3|r2 for x - (71 12)Т €...
3. Now suppose that (a,b), (a2, b2),..., (aq, be) are l distinct points on R2. Let X be the set formed by these l points. Prove that there are l vector fields F1, F2,..., Fe, each defined on R2X (the set R2 without the points in X), with the following properties: (i) curl F; = 0 on RP X for all i = 1, ..., l. (ii) (“linearly independent”) If C1,C2, ..., Ce are real numbers such that the vector...
3. (5 points) Chapter 3. #4 (modified). Prove the following properties related to Hermitian operators: (a) If Ô and 6 are Hermitian, so is Ê + 0. (b) If z is any complex number and if Ô is Hermitian, then zÔ is Hermitian if and only if z is real. (c) If Ê and Ộ are Hermitian and if they commute, the Ộ Ô is Hermitian. In your proof, indicate explicitly which step requires the two operators to commute. (d)...