Question: Let V be a normed vector space. Prove that a linear map L: V V defines an isometry of V for the given norm

Let V be a normed vector space. Prove that a linear map L: V → V defines an isometry of V for the given norm if and only if it maps the unit sphere S1 = {||u|| = 1} to itself:
L[S1] = {L[u] | u ∈ S1} = S1.

Step by Step Solution

★★★★★

3.44 Rating (176 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

First if L is an isometry and u 1 then Lu ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

952-M-L-A-E (2505).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let V be a vector space with subspaces U and W. Prove multiplication that U W is a subspace of V.

Let V be a vector space with subspaces U and W. Give an example with v = [R2 to show that U U W need not be a subspace of V.

Let V be a vector space with subspaces U and W. Define the s u m of U a n d W to be U + W = {u + w: u is in U, w is in W} (a) If V = R3, U is the x-axis, and W is the y-axis, what is U + W? (b) If U...

,, kindly solve Consider the Linear Regression Model 1. For any X = x, let Y = x8 + U, where B E R*. 2. X is exogenous. 3. The probability model is { f (u; ) is a distribution on R : E, [U] = 0, Var,...

Solve in details The symbol K will denote either R or (C. Exercises If V is a vector space over K, a linear functional on V is a linear map (,0: V > K. The set of linear functionals from V to K is...

,. solve the following Consider the following models describing the relationship between monthly return on the Standard & Poor's 500 stock market index (rsp500), the re- turn on three-month T-bills...

1 Let V and W be vector spaces over F with V finite-dimensional, and let U be any subspace of V. Given a linear map SE L(U,W), prove that there exists a linear map TE L(V, W) such that, for every u e...

Please solve in details The symbol K will denote either R or (C. Exercises If V is a vector space over K, a linear functional on V is a linear map (,0: V > K. The set of linear functionals from V to...

9.5.6 Kernel Principal Component Analysis (KPCA) As discussed above, RKHS allows us to delinearize some linear algorithms. We give an example here with the Principal Component Analysis (see Exercise...

Let A = [a;l E Fxn be a square n x n matrix. We define the trace of A to be the number Tr(A) = akk- k=1 a) Prove that Tr : FXn F is a linear function. b) Let A, B e F"X". Prove that Tr(AB) = Tr(BA)....

As a means of enhancing heat transfer from high-performance logic chips, it is common to attach a heat sink to the chip surface in order to increase the surface area available for convection heat...

Can I have a help for a Laurent Expansion for the following? I need it on summation notation using the general formula below. I want it done on this way If it cannot be carried out that way, one can...

Assume that Amazon.com sells the MacBook Pro, a computer produced by Apple, for a retail price of $ 1 , 4 1 0 . Amazon arranges its operations such that customers receive products directly from Apple...

A student makes the following argument: A price floor reduces the amount of a product that consumers buy because it keeps the price above the competitive market equilibrium. A price ceiling, on the...

The tar content in 30 samples of cigar tobacco follows: (a) Is there evidence to support the assumption that the tar content is normally distributed? (b) Find a 99% CI on the mean tar content. (c)...

A manufacturer of electronic calculators takes a random sample of 1200 calculators and finds that there are eight defective units. (a) Construct a 95% confidence interval on the population...

An article in The Engineer (Redesign for Suspect Wiring, June 1990) reported the results of an investigation into wiring errors on commercial transport aircraft that may produce faulty information to...

Need help with D Below: Consolidated Pasta is currently expected to pay annual dividends of $10 a share in perpetuity on the 2.7 million shares that are outstanding. Shareholders require a 10% rate...

At the time of her grandson's birth, a grandmother deposits $1000 in an account that pays 8% compounded monthly. What will be the value of the account at the child's twenty-first birthday, assuming...

QUESTIONS REIT Inc. paid a dividend of 52 per share per year on its common stock in past 10 years. And it is expected to continue paying two dollars a share long in future. From the info, one share...