Are the following proofs correct(3)?

Question

A list of some questions from class and my attempt at them are given below. Would appreciate any advice on what I did wrong or on how to attempt some of the questions.

5. The set $C^∞(R^2)$ of functions $f(x, y)$ of two variables, which are infinitely many times differentiable with respect to both variables $x$ and $y$ is a linear space (prove this). The Laplace operator $∆: C^∞(R^2) → C^∞(R^2)$ is a mapping which to a function $f(x, y)$ assigns the function $∂^2f/∂x^2 + ∂^2f/∂y^2$ . Show that the Laplace operator is a linear operator.

My attempt:

First showing $C^∞(R^2)$ is a linear space with the 8 linear space axioms, let $f_n , f_m, f_o, O,I \in C^∞(R^2)$

1. $f_n+ f_m = f_m+ f_n$
2. $(f_n+ f_o)+f_3 = f_n+ (f_m+f_o)$
3. $f_n + O = f_n$
4. $f_n+f_m = O$, for any element $f_n$ there exist some element $f_m$
5. $\alpha(f_n + f_m) = \alpha f_n + \alpha f_m$
6. $(\alpha + \beta)f_n = \alpha f_n + \beta f_n$
7. $(\alpha\beta)f_n = \alpha(\beta f_n)$
8. $If_n = f_n$

No idea how to show these axiom.. tips on how to start?

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6.Let X be a linear space (over R). The set R is also a linear space, and hence, one can consider linear operators which act from X to R. Such linear operators are usually called linear functionals.

(a) Show that the definite integral I : C[a, b] → R, which to a continuous function f on [a, b] assigns its integral R b a f(x) dx, is a linear functional.

(b) Show that a mapping δ : C(R) → R, which to a continuous function f on R assigns its value at 0, is a linear functional.

My attempt:

Other then knowing that a functional is a mapping of functions to constants in the field, I have no idea how to start this.

Answer 1

Regarding problem 5:

• What's the simplest thing "$+$" could mean when you start with two functions defined on the same domain? In fact, how is $(f+g)(x,y)$ defined (in basic algebra)? Pointwise! Is pointwise addition commutative and associative?
• If addition is pointwise, what is the only function that could be the identity under addition? Is that function $C^\infty$?
• Given a function $f$, what function do I add to it pointwise to get the additive identity? Is that thing at least as smooth as $f$?
• How do we scalar multiply functions? Hint: $f+f = 2f$.
• If we scalar multiply, is the function still smooth enough?
• Note that these scalar multiplications are the same as pointwise multiplications by constant functions. Is there an obvious function to be the multiplicative identity? Is that function smooth enough?

Really, what you are checking is that the operations are well-defined and the set $C^\infty(\mathbb{R}^2)$ is closed under these operations.

Regarding problem 6: These are pointwise operations again. Are addition, the zero, the one, and constant multiplication continuous? Are the continuous functions closed under the operations?