a) Prove that given f C[a, b], then there is a sequence of one-variable polynomials Pn

Question:

a) Prove that given f ∈ C[a, b], then there is a sequence of one-variable polynomials Pn such that Pn → f uniformly on [a, b] as n → ∞.
b) Prove that the metric space C[a, b] (see Example 10.6) is separable.

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

Step by Step Answer:

Related Book For book-img-for-question

An Introduction to Analysis

ISBN: 978-0132296380

4th edition

Authors: William R. Wade

See More Books

Question Posted: Mar 29, 2016 04:18 AM

See More Questions

Let R = [a, b] Ã [c, d] be a rectangle in R2. A function f is said to have separated variables if for some scalars ck and functions fk C[a, b], gk C[c, d]. Prove that given f C(R) there is a...
Let T = [0, 2). a) Prove that the function (x) := (cos x, sin x) is 1-1 from T onto B1(0, 0) R2. b) Prove that p(x, y) := || (x) - (y)|| is a metric on T. c) Prove that a function f is continuous on...
Suppose that (a, b) is an open interval, that f: (a, b) R is differentiable on (a, b), and that x0 (a, b) is a proper local maximum of f (see Exercise 4.1.8). a) Prove that given > 0, there exist...
Reid Corporation's balance sheet at January 1, 20X9 reflected the following balances: Cash & Receivables $ 30,000 Inventory $ 75,000 Land $125,00 Building & Equipment (net) $850,000 Common Stock...
What are ADRs?
1. Indicate whether the following statements are true or false: a. Managerial accounting information is designed primarily to meet the needs of external users such as shareholders, creditors, and the...
Let $y_{i j}$ for $i=1, \ldots, b$ and $j=1, \ldots, k$ be the measurement for the unit assigned to the $j^{\text {th }}$ treatment in the $i^{t h}$ block. a. Show that \[S S_{T}=S S_{\text {Treat...
Students in the ACHANCHA survey were asked Within the last 12 months, has stress negatively affected your academics? Figure 2.4(a) shows a segmented bar chart for response frequencies while Figure...
Based on the following information, compute cash flows from financing activities. Purchase of short-term investments Dividends paid Interest paid Additional short-term borrowing from bank $ 630 890...
Turner Excavation maintains a checking account and has decided to open a petty cash fund. The following petty cash fund transactions occurred during July: July 2 Established a petty cash fund by...
Suppose that X is connected. Prove that if there is a nonconstant, continuous function f: X R, then X has uncountably many points.
A polynomial on Rn is a function of the form where the aj1,. . . . . .. j n 's are scalars and N1,...,Nn N. Prove that if A is compact in Rn and f C(A), then there is a sequence of polynomials Pk on...
News Story 16: More on TV Violence.
During attenuation, when tryptophan levels are high, the ________ stem-loop forms and transcription _________ the trpL gene. a. 12, ends just past b. 34, ends just past c. 12, continues beyond d. 34,...
Lets suppose a researcher mixed together nucleotides with the following percentages of bases: 30% G, 30% C, and 40% A. If RNA was made via polynucleotide phosphorylase, what percentage of the codons...
Scientists propose that the first macromolecules in protobionts were a. DNA molecules. b. RNA molecules. c. proteins. d. all of the above.
An example of a posttranslational covalent modification that may affect protein function is a. phosphorylation. b. acetylation. c. methylation. d. Any of the above can affect protein function.
A form of chromatin that silences gene expression and varies in location among different types of cells is a. euchromatin. b. constitutive heterochromatin. c. facultative heterochromain. d. Both b...
The cross section of a transmission line is shown in Figure 14.68. Use the finite difference method to compute the characteristic impedance of the line. 8 = 80 1 cm 3 cm 1 cm 3 cm
Refer to the data in QS 10-1. Based on financial considerations alone, should Helix accept this order at the special price? Explain.
Exercise 11-7 gave 20 observations on y = highway gasoline mileage and x = engine displacement. (a) Test for significance of regression using a = 0.01. Find the P-value for this test. What...
Exercise 11-8 gave 13 observations on y = green liquor Na2S concentration and x = production in a paper mill. (a) Test for significance of regression using a = 0.05. Find the P-value for this test....
Exercise 11-9 presented data on y = blood pressure rise and x = sound pressure level. (a) Test for significance of regression using a = 0.05. What is the P-value for this test? (b) Estimate the...
Use the extension of set-builder notation concepts and the roster method to write the given set. D= {2x-91xE N and x < 5}
Statement of financial position/Balance sheet structure Level of difficulty: Moderate Chugoku Electric Power Company, Inc., was established in 1951 as one of ten electric power companies in Japan. It...
60.Blackstone Company uses a process costing system and has 14,000 units remaining in the work-in-process inventory. These units are 75% and 40% complete with regard to materials and conversions...

a) Prove that given f C[a, b], then there is a sequence of one-variable polynomials Pn

Question:

Step by Step Answer:

a Clearly the collection P of polynomials on R is an algebra in Ca b that contains the ...View the full answer

An Introduction to Analysis

Students also viewed these Numerical Analysis questions