Functional Analysis Answers

Prove that

i) 𝑑 𝑎𝑥 , 𝑎𝑦 = 𝑎 d(x,y)

ii)𝑑 𝑎 + 𝑥 , 𝑎 + 𝑦 = 𝑑 𝑥 , 𝑦

where d is a metric induced by on a normed space X

Proof whether the following operations are inner product operations:

⟨x, y⟩ = 2x₁y₁ − x₁y₂ − x₂y₁ + 2x₂y₂, x=(x₁, x₂), y=(y₁, y₂)

Show that the space L[a,b] of all square integrable functions on the interval [a,b] is a linear space over a vector field R

Show that the space L²(R) of all square of summable real sequence is a linear space over a vector field R

Show that the space L(R) of all bounded real sequence is a linear space over a vector field R

Show that the space B[a,b] of all bounded real value functions on the interval [a,b] is a linear space over a vector field R

Show that the space R[a,b] of all Riemann integrable functions on the interval [a,b] is a linear space over a vector field R

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

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]