Fornal define real projective space as the quotient space IRP",- (Rut\{0})/~ where (20,..., an ) ~(bg.-,...
Let S2 denote the
2-dimensional sphere. Define the complex projective line
1 as the quotient space
2 \ {0} / ∼ , where ∼ is the equivalence relation on
2 \ {0} that x ∼ y if x = λy for some λ∈C. Prove that
S2 and
1 are homeomorphic.
Let S denote the 2-dimensional sphere. Define the complex projective line CP as the quotient space C {0}/~, where is the equivalence relation on {0} that I ~y if r...
Let V = R3[x] be the vector
space of all polynomials with real coefficients and degress not
exceeding 3.
Let V-R3r] be the vector space of all polynomials with real coefficients and degress not exceeding 3. For 0Sn 3, define the maps dn p(x)HP(x) do where we adopt the convention thatp(x). Also define f V -V to be the linear map dro (a) Show that for O S n 3, T, is in the dual space V (b) LetTOs Show...
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...
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...
Real analysis
10 11 12 13 please
(r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
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Budgetary Policy and Economic Growth Errol D'Souza The share of capital expenditures in government expenditures has been slipping and the tax reforms have not yet improved the income...