Functional Analysis Answers

1. . Under a set of controlled laboratory conditions, the size of the population P of a certain bacteria culture at time t (in seconds) is given by the function P(t)= 3t²+ 3e^t+10, (t greater than/ equal to 0)

(i) What is the size of the population after 1 minute?

(ii) Find the average rate of change of P at t = 2 and t = 6?

(iii) How fast is the size of the population changing after 1 minute?

[Verify your answer by MATHEMATICA and attach the printout of the commands and output]

Describe verbally how to solve y=mx+c. What assumptions have you made about the value

of ?

If a third-degree polynomial has a lone x-intercept at x=a , discuss what this implies about the linear and quadratic factors of that polynomial

Kindly answer this as soon as possible. Urgent Elaborate each step.

Show that Euclidean space and unitary space are not compact. Explain each step. 

If (x_n) and (y_n) are sequences in the same normed space X, show that x_n→x and y_n→y implies x_n+y_n→x+y as well as αx_n→αx, where α is any scalar.

Let X and Y be normed spaces, T∈B(X,Y) and (x_n) a sequence in X. If x_n→x₀, show that Tx_n→Tx₀.

If x_n∈C[a,b] and x_n→x∈C[a,b]. Show that (x_n) is pointwise convergent on [a,b], that is, (x_n(t)) converges for every t∈C[a,b].