Intereting Posts

Proving Set Operations
The Gradient as a Row vs. Column Vector
Continued fraction of $e^{-2\pi n}$
Proof for convergence of a given progression $a_n := n^n / n!$
Isomorphism of $\mathbb{Z}/n \mathbb{Z}$ and $\mathbb{Z}_n$
Agreement of $q$-expansion of modular forms
Reference for Fredholm Integral Equations..
If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable
Heuristic\iterated construction of the Weierstrass nowhere differentiable function.
In category theory: What is a fibration and why should I care about them?
Inertia group modulo $Q^2$
Compact operators, injectivity and closed range
Infinite sum and the conditions with tanh
Norm inequality for sum and difference of positive-definite matrices
generic regularity of affine varieties

Is the subset consisting of all integrable (or square integrable) smooth functions of the set of all integrable (or square integrable) functions, dense under the usual Euclidean or integral of absolute difference metric?

By smooth I mean derivatives of all orders exist.

- Finding $f(x)$ from a functional equation
- Prove that $f]] = f$
- Solving a radical equation for real $x$
- How do you calculate this sum $\sum_{n=1}^\infty nx^n$?
- how can we convert sin function into continued fraction?
- Function that sends $1,2,3,4$ to $0,1,1,0$ respectively

- minimum and maximum problem
- Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$
- Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$
- Can it be that $f$ and $g$ are everywhere continuous but nowhere differentiable but that $f \circ g$ is differentiable?
- Why does an open interval NOT have measure zero?
- The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$)
- Norm for pointwise convergence
- Construction of a continuous function which is not bounded on given interval.
- What's the arc length of an implicit function?
- Finit Sum $\sum_{i=1}^{100}i^8-2 i^2$

Yes. In fact, by the Stone-Weierstrass theorem and the existence of smooth bump functions, smooth functions with compact support are uniformly dense in the space of continuous functions with compact support. Uniform density implies $L^2$ and $L^1$ density for functions with compact (and therefore finite measure) support, and since continuous functions with compact support are dense in $L^2$ and $L^1$, the result follows.

If you wanted to see this more directly, you can go through the iterations of approximating an arbitrary ($L^1$ or $L^2$) function with a bounded function with bounded support, then with a simple function, then with a step function, and finally approximate the step function with a smooth function using bump functions.

Jonas’s argument is good. Another proof is: given $f \in L^p$ (here $p=1,2$), take the convolution of $f$ with a sequence of mollifiers $\eta_\epsilon$. Using properties of convolutions, it’s easy to check that $f * \eta_\epsilon$ is a smooth function, and that $f * \eta_\epsilon \to f$ in $L^p$ as $\epsilon \to 0$. This has the advantage of being a little more direct.

**Edit:** For a reference, see Folland’s *Real Analysis*, section 8.2.

The smoothness of $f * \eta_\epsilon$ is Proposition 8.10 and comes from differentiating under the integral sign in the convolution (with justification!), and choosing to put the derivative on $\eta_\epsilon$. Intuitively, it comes from the idea that convolution is an “averaging” operation and tends to smooth, smear, or blur rough areas of $f$ together, and so should be a smoothing operation. (The wikipedia article has a nice animation illustrating this.)

The fact that $f * \eta_\epsilon \to f$ in $L^p$ is Folland’s Theorem 8.14 (a), and it’s pretty elementary. He also has Proposition 8.17 which proves that $C^\infty_c$ is dense in $L^p$, but it sort of inexplicably starts by using the fact that $C_c$ is dense in $L^p$. I suppose this is used to get a compactly supported function, so that you can approximate $f \in L^p$ by functions which are not only smooth (which $f * \eta_\epsilon$ is) but also compactly supported (which $f * \eta_\epsilon$ need not be, although $\eta_\epsilon$ is). But an easier argument would be to first approximate $f$ in $L^p$ norm by a function $g$ which is compactly supported but not necessarily continuous; for example, $g = f 1_{[-N,N]}$ for large $N$ (this works by dominated convergence), and then apply mollifiers to $g$. Unless, of course, there is some subtlety that I’ve missed.

**Edit 2**: Indeed there is. Folland’s 8.14 (a) relies upon the fact that translation is strongly continuous in $L^p$, which uses the density of $C_c$. So apparently it is not so easy to bypass this step, and that destroys a lot of the “directness” of my argument.

- Mapping circles using Möbius transformations.
- Integral of $\sqrt{1-x^2}$ using integration by parts
- Yablo's paradox? a paradox without self-reference
- Solving Pell's equation(or any other diophantine equation) through modular arithmetic.
- To what extent are the Jordan-Chevalley and Levi Decompositions compatible.
- Calculate the p.m.f. of a non-monotonous function of a random variable
- How can $\lim_{n\to \infty} (3^n + 4^n)^{1/n} = 4$?
- Is this proof that $\sqrt 2$ is irrational correct?
- How would you solve the diophantine $x^4+y^4=2z^2$
- Calculate $\mathbb{E}(W_t^k)$ for a Brownian motion $(W_t)_{t \geq0}$ using Itô's Lemma
- Grassman formula for vector space dimensions
- Elementary question about Cayley Hamilton theorem and Zariski topology
- If $f$ continuous differentiable and $f'(r) < 1,$ then $x'=f(x/t)$ has no other solution tangent at zero to $\phi(t)=rt$
- Rewriting repeated integer division with multiplication
- Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$