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If I have to refer to a function $f$ without defining it explicitly, what should I write? Or is there no notation I can use but explicitly defining $f$?

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    $\begingroup$ I'm not sure why you have a problem here - it is common in advanced math to write about functions you never explicitly defined. It comes with the territory. $\endgroup$ Aug 12, 2021 at 20:02
  • $\begingroup$ dear my2cents, do you know what the domain and codomain of the function are going to be? if so, (suppose they are $X$ and $Y$, respectively) then you can just say "let $f:X\to Y$ be a function", or "let $f$ be a function from $X$ to $Y$" $\endgroup$ Aug 12, 2021 at 20:04
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    $\begingroup$ ahhh, I see; I personally think that it is best for clarity to write something like "for each $i\leqslant n$, define $f_i:\mathbb{R}\to\mathbb{R}$ by $f_i(x)=e^{\lambda_i x}$." but ultimately it's up to you; what you've written in that sentence seems quite clear already to me, although maybe others would disagree $\endgroup$ Aug 12, 2021 at 20:22
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    $\begingroup$ And then after, if I need to refer to $f_i$, I can just write $f_i$ (e.g. $Tf_i$, which is in my next sentence). $\endgroup$
    – my2cents
    Aug 12, 2021 at 20:26
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    $\begingroup$ You could also use a notation like $e^{\lambda(\cdot)}$ or $x \mapsto e^{\lambda x}$ for this purpose. However I think just defining $f_i$ explicitly as @atticus says would probably be best if you want to refer to them later. You could also use a notation like $f_{\lambda}(x) = e^{\lambda x}$ if you don't want as many indices. $\endgroup$ Aug 12, 2021 at 20:40

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I think what you’re looking for is the notation $x \mapsto x^2$ which allows you to write the function $f$ where $f(x) = x^2$ without giving it a name. (This notation also leaves out the domain and codomain which might be fine if they’re clear from context or if you mention them in the surrounding text.)

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It seems like writing "Let $f: V\to W$ be a function defined by [formula]" or similar is clearest.

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