Give an example (without proof) of each of the following:
An integrable function \(f: \mathbb{R} \to \mathbb{R}\) so that the function \(F(x) = \int_0^x f(t)\,dt\) is differentiable everywhere but at \(x = 1\).
A sequence \(\{a_n\}\) of real numbers such that \(\sum_{n=1}^{\infty} a_n\) converges and \(\sum_{n=1}^{\infty} a_n^2\) diverges.
A basis for the subspace of \(\mathcal{P}_3\) spanned by \(x^2 + x + 1\), \(x^3 - x + 2\), \(x^3 + x^2 + 3\), and \(-x^3 + x^2 + 2x + 1\).
Solution
Take \[ f(t)=\begin{cases}0,&t<1,\\1,&t\ge1.\end{cases} \] Then \[ F(x)=\int_0^x f(t)\,dt \] is differentiable for every \(x\ne1\) and not differentiable at \(x=1\).
Take \[ a_n=\frac{(-1)^n}{\sqrt n}. \] The alternating-series test gives convergence of \(\sum a_n\), while \[ \sum a_n^2=\sum\frac1n \] diverges.
Relative to the coefficient basis \((1,x,x^2,x^3)\), the four given polynomials have columns \[ (1,1,1,0),\ (2,-1,0,1),\ (3,0,1,1),\ (1,2,1,-1). \] The first, second, and fourth are linearly independent, and the span of all four has rank \(3\). Hence one valid basis is \[ \boxed{\{x^2+x+1,\ x^3-x+2,\ -x^3+x^2+2x+1\}}. \]