Let (xm) be a Cauchy sequence in a normed linear space x of dimension n. Let {x1,x2,...,xn}

Question:

Let (xm) be a Cauchy sequence in a normed linear space x of dimension n. Let {x1,x2,...,xn} be a basis for X. Each term x''' has a unique representation

1. Using lemma 1.1, show that each sequence of scalars xmi is a Cauchy sequence in „œ and hence converges to some Î±i ˆˆ „œ.
2. Define

Show that x ˆˆ X and that xm †’ x.
3. Conclude that every finite-dimensional normed linear space is complete.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Foundations of Mathematical Economics

ISBN: 978-0262531924

1st edition

Authors: Michael Carter

See More Books

Question Posted: May 02, 2016 11:30 AM

See More Questions

Let f: X Y be uniformly continuous. If (xn) is a Cauchy sequence in X,(f(xn) is a Cauchy sequence in Y.
Let x0 be an element of a normed linear space X. There exists a linear functional f X* such that ||f|| = 1 and f(x0) = ||x0||.
Let S be a nonempty set in a normed linear space X. Show 1. S* is a closed convex cone in X* 2. S** is a closed convex cone in X 3. S S**
Payroll Assignment - (50 Marks) The following employees are working in the ABC Clinic, they are paid biweekly. Calculate the gross income, net income, and the total remittance that ABC is to report...
A, a middle-level manager of Drug Company X, has been told by her boss, B, to change some figures on the percentage of rats that died as a result of injections of a new drug for which Company X is...
A software development project at day 70 exhibits an actual cost of $78,000 and a scheduled cost of $84,000. The software manager estimates a value completed of $81,000. What are the cost and...
Consider the following cases. (a) A small 0.6-in.-long fish swims with a speed of $0.8 \mathrm{in} / \mathrm{s}$. Would a boundary layer type flow be developed along the sides of the fish? Explain....
The adjusted trial balance for Rochester Electronics, Inc. on November 30, the end of its first month of operation, is as follows: The general ledger reveals the following additional data: a. There...
A venturi meter is used for measuring flow of water along a pipe.The diameter of the Venturi throat is two fifths the diameter of the pipe.The inlet and throat are connected by water filled tubes to...
a. To simplify the analysis, assume for now that there is virtually no turnaround time between flights so the next flight can begin as soon as the current flight ends. (If an immediate next flight is...
To prove lemma 1.1, assume, to the contrary, that for every c > 0 there exists x lin{x1; x2; . . . , xn} such that where x = a1x1 + a2x2 + ¢ ¢ ¢ +anxn. Show that this implies that 1....
Two norms ||x||a and ||x||b on a linear space are equivalent if there are positive numbers A and B such that for all x X, The following exercise shows that there essentially only one finite...
Information for two companies in the same industry, Wingate Corporation, is presented here. Instructions Compute free cash flow for both companies and compare. Merrill Corporation Corporation Wingate...
Absolute ownership of real property. a. accession b. adverse pos session c. in causa mortis d. eminent doma in e. fee s imple f. freehold e state g. grantee h. inter vivos i. joint te nancy j....
Indicate whether each of the following statements is true or false by writing T or F i n t he a nswer c olumn. The United States lacks the abundance of natural resources that many other countries...
A document of transfer of real property. a. accession b. adverse pos session c. in causa mortis d. eminent doma in e. fee s imple f. freehold e state g. grantee h. inter vivos i. joint te nancy j....
Prism Co acquired 80% of the stock of Sapphire Co for $300,000 on 1 January 20x7. At acquisition date, Sapphire reported retained earnings of $150,000. The excess of Prism Cos acquisition cost over...
A gift given in anticipation of death. a. accession b. adverse pos session c. in causa mortis d. eminent doma in e. fee s imple f. freehold e state g. grantee h. inter vivos i. joint te nancy j....
Ralph placed a classified ad to sell his used SUV for $18,500. After 2 weeks, he didnt sell the SUV, and the newspaper suggested lowering the price 5%. What would the new price be if Ralph reduced it...
Juarez worked for Westarz Homes at construction sites for five years. Bever was a superintendent at construction sites, supervising subcontractors and moving trash from sites to landfills. He...
Two partners have decided to sell the manufacturing business they have been running. They have lined up five prospective buyers and have hired a consultant to help them with the bidding and the sale....
A sealed-bid auction is going to be held in the near future for Medex Pharmaceuticals. You represent one of two companies expected to enter bids. The basic data available to your firm consist of an...
A given stock is currently priced at $100. Historically, its annual return has been 12 percent with a standard deviation of 15 percent. Build a spreadsheet simulation model for the stock price, using...
4.5. Multiplication in GF(24): Compute A(x)-B(x) mod P(x) in GF(24) using the irreducible polynomial P(x) = x++x+1. What is the influence of the choice of the reduction polynomial on the computation?...
Write a C++ program that calculates the volume of a cylinder. Define appropriate variables for the diameter and the height of the cylinder. Assuming the units of these variables are in inches,...
a. Show the d and values that result from running breadth-first search on the directed graph of Fig 1, using vertex A as the source. (Ascending Order) b. Show the d, f and values that result from...

Let (xm) be a Cauchy sequence in a normed linear space x of dimension n. Let {x1,x2,...,xn}

Question:

" 3 2"х, + х , т. х, +

Step by Step Answer:

1 Let be a normed linear space of dimension and let x 1 x 2 x be a basis for Let ...View the full answer

Foundations of Mathematical Economics

Students also viewed these Numerical Analysis questions