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In other words, given a countable sequence of neighborhoods of $f(x)=0$, how to construct another open neighborhood that doesn't contain any of these neighborhoods? Thanks.

Robert Cardona
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a non-normal user
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1 Answers1

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Denote by $B_m$ the open ball in $R^d$ of radius $m$, and let $\bar B_m$ denote the closed ball of radius $m$. Given a countable collection of non-empty open neighborhoods of $f=0$, say $U_n$, fix, for each $n$, a non-zero function $f_n\in U_n$ such that $f_n$ has support in $\bar B_{m_{n+1}}\setminus B_{m_n}$, where $m_n$ is a strictly increasing sequence, picked inductively large enough so that such a function $f_n$ exists. Pick also $x_n\in \bar B_{m_{n+1}}\setminus B_{m_n}$ with $f_n(x_n)\neq 0$. The following set is open in the space of smooth compactly supported functions (see page 152 of the book 'functional analysis' by rudin) $$ U=\{\phi\in C_c^\infty \,| \; |\phi(x_n)|<|f_n(x_n)|\}. $$ Since $f_n\notin U$, no $U_n$ is contained in $U$.

Jonas Dahlbæk
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  • Thanks! Totally random but related question, what exactly is an open ball in semi-norm topology (smooth topology) of smooth function with support in a compact subset (not Rd) – a non-normal user Oct 23 '14 at 20:09
  • I would not use the term 'open ball' unless there is some metric which gives rise to the topology. The particular topology on smooth functions of compact support is somewhat involved to write down. It is described on wikipedia (this is the same description that is given in the book by Rudin). I remember seeing a nice, concrete definition in the book 'The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis' by Hörmander, but I don't have that book with me currently. – Jonas Dahlbæk Oct 23 '14 at 20:19
  • So I guess for smooth function with compact support in a compact subset of Rd, the topology can be described by the metric in Fréchet space equivalently? – a non-normal user Oct 23 '14 at 20:34
  • No, that space is not complete, since you can have smooth compactly supported functions that converge uniformly on compact subsets to a constant function (which does not have compact support). – Jonas Dahlbæk Oct 23 '14 at 20:44
  • I think you can write down a family of seminorms in the form $$p_m(f)=\max \sum_{|\alpha|<m}\rho|\partial^\alpha f|,$$ where $\rho$ is a positive function with the property that for each compact set $K$ there is an $\epsilon>0$ such that $\rho\geq\epsilon$ on $K$. I'm not sure that my memory serves me well though. Check out the book by Hörmander if you want details. – Jonas Dahlbæk Oct 23 '14 at 20:47
  • Thank you very much. I will go ahead check out that book. – a non-normal user Oct 23 '14 at 20:49