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Functional Analysis Answers

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Q. Which of the following sets are countable or uncountable.
(i) set of real numbers
(ii) set of complex numbers
Q. Which of the following sets are countable or uncountable.
(i)set of negative integers
(ii) set of rational numbers
Q. Which of the following sets are countable or uncountable.
(i) set of natural numbers
(ii) set of whole numbers
(iii) set of positive integers
show that the absolute value of a linear functional has the subadditive and positive homogeneous properties.
show that a norm on a vector space X is a sublinear functional on X
Q.1: Find the fourier series of f(x)=│x│ on [-π, π]
Q.2: Find the fourier series of f(x)=2-x2 on (-2<x<2)
1
limx→1(1+x+x2+……...+xm−1)
= …….
1
m-1
m
-1
if f:X to Y and g:Y to Z then the domain of gof is X and co-domain is Z , what is the domain and co-domain of fog, fof, gog, here in this case?
Just wanted to include a tad more information**

You are to design a real life object using at least 5 different types of functions. There are an infinite number of equations that will produce a shape that fulfils the design criteria, therefore uniqueness will be taken into consideration.

Your task is to design a UNIQUE real-life object using at least 5 different mathematical functions or relations. You can choose from any of the following (eg. linear, parabolic, hyperbolic, exponential, logarithmic, trigonometric, circles) that would be appropriate and accurate for your object.

You are to:
1. Submit a list of functions or relations that were used to make up your design . Each function must have accurate domains
2. Produce an accurate graph of your design from GRAPHmatica. This will justify the reasonableness of your results.
3. Include all working (rough drafts) for the development of your design
4. Fully justify your choice of each function and its appropriateness to the design
You are to design a real life object using at least 5 different types of functions. There are an infinite number of equations that will produce a shape that fulfils the design criteria, therefore uniqueness will be taken into consideration.

Your task is to design a UNIQUE real-life object using at least 5 different mathematical functions or relations. You can choose from any of the following (eg. linear, parabolic, hyperbolic, exponential, logarithmic, trigonometric, circles) that would be appropriate and accurate for your object.

This is the last question on my math assignment (Introduction to functions) and I just cant figure out how to do it. Some help would be deeply appreciated