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(g) Prove that Rn - {0} and sn-1 X R are homeomorphic spaces. (Hint: Consider the...
We used definition of homeomorphic as follows.
If X and Y are topological spaces, a function f: X to Y is
called homeomorphism if
1. f is continuous
2. f is bijective
3. inverse of f is continuous
And in this case, we say that X is homeomorphic with Y.
Thank you !
infinite) (5) Prove that all semiopen intervals in R (finite or homeomorphic are
infinite) (5) Prove that all semiopen intervals in R (finite or homeomorphic are
Let a continuously differentiable function f: Rn → R and a point x E Rn be given. For d E Rn we define Prove the following statements: (i) If f is convex and gd has a local minimum at t-0 for every d E R", then x is a minimiser of f. (ii) In general, the statement in (i) does not hold without assuming f to be convex. Hint: For) consider the function f: R2-»R given by
Let a continuously...
4. Prove the following statement: Consider the ODE x = f(x) with x : J C R → Rn and f : Rn → Rn. If a continuously differentiable real-valued function V = V(x) exists such that (a) V is defined on Bs(0) {x E Rn : Irl < δ} (b) V(x) 0 for x E Bs(0) 1 fo) (c) V 0) 1 (o then the origin is unstable. (x) >0 for rE Bs
4. Prove the following statement: Consider...
Question 8 (Chapters 6-7) 12+2+2+3+2+4+4-19 marks] Let 0メS C Rn and fix E S. For a E R consider the following optimization problem: (Pa) min a r, and define the set K(S,x*) := {a E Rn : x. is a solution of (PJ) (a) Prove that K(S,'). Hint: Check 0 (b) Prove that K(S, r*) is a cone. (c) Prove that K(S,) is convex d) Let S C S2 and fix eS. Prove that K(S2, ) cK(S, (e) Ifx. E...
Topology
3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t € [2 – 1, 2+1]}. Prove that g(x) is also continous. Hint: To prove g(x) is continous at x = xo. You can consider the continuity of f(x) at the two boundary point xo - 1 and xo +1. When x get close to xo, the points in (7 - 1, + 1) is close to xo - 1, xo + 1, or inside...
(c) [5 points] Prove that f(r) [5 p ) = Σ (-1-rn oints Prove that f(x converges uniformly on [-c, c when 0<c<1. lenny
Let f : Rn × Rn → R be the inner product function: f(r,y)-(2,3) 1. Using the definition of multivariable derivative, calculate D fab and the Jacobian matrix f'(a, b) 2. If f, g : Rn → R are differentiable and h : R → R is defined by h(t)-(f(t), g(t)), show that 3. If f : R → Rn is differentiable and Ilf(t)ll = 1 for all t, show that(f,(t)T,f(t))-0
Assume f : R" → R is twice continuously differentiable. Prove that the following are equivalent: (a) f(ex + (1-8)ì) < ef(x) + (1-8)/(x) for all x, x E Rn and 0 < θ < 1 (b) f(x)+ /f(x) . (x-x) -f(r) for all x,x E R" (c) f(x) > 0 for all x E R" Hint: Look at : RRdefine by gt) f(x + ty) where x, y E R. First show g is convex (as a function of...
Please prove
Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function f: (a, b)- (c, d) for which f(a)-c and f(b)-d and show this is a homeomorphism.)
Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function...