(a) The volume of a regular pyramid.

Question

(1) Let R be a field of real numbers and X=R³, X is a vector space over R, let M={(a,b,c)/ a,b,cE R,a+b=3-c}, show that whether M is a hyperplane of X or not (not by definition). متکاری Xn-XKE 11Xn- Xmit (2) Show that every converge sequence in a normed space is Cauchy sequence but the converse need not to be true. EK 2x7 (3) Write the definition of continuous map between two normed spaces and write with prove the equivalent statement to definition. (4) Let be a subset of a normed space X over a field F, show that A is bounded set iff for any sequence in A and any sequence in F converge to zero the sequence converge to zero in F. އ

Answer 1

Question

Chapter 63, Problem 14A

To determine

(a)

The volume of a regular pyramid.

To determine

(b)

The weight of one cubic inch of cast iron.

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(1) Let R be a field of real numbers and X=R³, X is a vector space over R, let M={(a,b,c)/ a,b,cE R,a+b=3-c}, show that whether M is a hyperplane of X or not (not by definition). متکاری Xn-XKE 11Xn- Xmit (2) Show that every converge sequence in a normed space is Cauchy sequence but the converse need not to be true. EK 2x7 (3) Write the definition of continuous map between two normed spaces and write with prove the equivalent statement to definition. (4) Let be a subset of a normed space X over a field F, show that A is bounded set iff for any sequence in A and any sequence in F converge to zero the sequence converge to zero in F. އ

Consider y (t) — y" (t) − y' (t) + y(t) = 0 (a) Denote new variables x1(t) := y(t), x2(t) := y' (t), x3(t) = y"(t) and solve the following system 0 1 0 x1(t) X' (t) = 0 1 X(t), X(t) = x2(t) -1 1 1 x3(t) = y(t) y' (t) y" (t) (b) Use your solution to the system to find the solution to the original equation (verify!).

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