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    A $C^\infty$ function vanishing on $(-\infty,0]$ but not on $(0,\infty)$, an infinite-dimensional vector space over $\ZZ/2\ZZ$, a non-diagonalizable matrix in $M_2(\CC)$, and a power series with radius of convergence $0$

    P-CQNPO

    Area
    Prelim
    Topics
    Counterexamples, Differentiation, Series of Functions
    Seen at
    UGA
    Years
    2007
    Status
    draft

    Give an example (proof not required) of each of the following:

    1. A function \(f:\mathbb{R}\to\mathbb{R}\) which satisfies \(f(x)=0\) for all \(x\le 0\), but which is nonzero for \(x>0\) and has derivatives of all orders for all \(x\).

    2. An infinite-dimensional vector space over the field \(\mathbb{Z}/2\mathbb{Z}\).

    3. A matrix in \(M_2(\mathbb{C})\) which is not diagonalizable.

    4. A power series \(\sum_{n=0}^\infty a_n z^n \in \mathbb{C}[[z]]\) with radius of convergence \(0\).

    <1>2. For part (b), take the polynomial ring \[ (\mathbb Z/2\mathbb Z)[x] \] viewed as a vector space over \(\mathbb Z/2\mathbb Z\). Its basis \(1,x,x^2,\dots\) is infinite.

    <1>3. For part (c), take \[ \begin{pmatrix}0&1\\0&0\end{pmatrix}. \] It is not diagonalizable over \(\mathbb C\).

    <1>4. For part (d), take \[ \sum_{n=0}^\infty n!\,z^n. \] Its radius of convergence is \(0\).

    Solution

    1. For part (a), take\[ f(x)= \begin{cases} 0,&x\le0,\\ e^{-1/x},&x>0. \end{cases} \]This function is \(C^\infty\) on \(\mathbb R\), vanishes on \((-\infty,0]\), and is nonzero for every \(x>0\).

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