Let f,g [a,b]→R be bounded functions andP any partition of [a,b]. -> Show that Lp(f)+Lpg)s Lplf+g) and Up(f+g)sUpf) +Up(g). Hence conclude that if f and g are integrable, then so is f+g, and the Riemann integral of f+g is equal to the sum of the Riemann integrals of f and g

Recall the inner product (f,g) = S, f(x)g(x) dx defined on the vector space C([-]) of continuous (real-valued) functions f : [-, π] → R. Observe the following integral formulas, which are valid for natural numbers m, n > 1: • sin(ma) cos(nx) dx = 0 (0 cos(mx) cos(nx) dx = sin(mx) sin(nx) dx = ifm #n | if m=n 1. What do these formulas say about the angles between the vectors sin(n.), sin(m.), cos(n.) and cos(m.) in the space C([-])? What do say you about the lengths of these vectors? 2. Do you think these vectors form a basis for the space C([-,])? Why or why not?

Let f : [0, 1] → R be defined by f(x) : = x. Show that f ∈ R  [0, 1] and compute ∫^1_0(f) using the definition of the integral.

3. Let V denote the vector space of all functions f : R" → R, equipped with addition + : V × V → V defined via (ƒ+g)(x) = f(x)+ g(x), x ≤ R", and scalar multiplication : R × V → V defined via (\ · ƒ)(x) = \ƒ(x), ● x ER". Rº Now let W = {ƒ : R^ → R : ƒ(x) = ax + b for some a, b € R}, i.e. the space of all linear functions R" → R. (a) Find a basis for W. You should prove that it is indeed a basis.