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Is there a Standard or non-standard notation for swapping symbols? For example, my proof is identical for an arbitrary vector space $V$ over the field $\mathbb F$, however the original proof is written for $\mathbb R^n$.

I want to indicate using notation to swap every instance of $\mathbb R^n$ (as a symbol) for $V$, (as a symbol), and the proof is syntactically equivalent.

I was thinking $\mathbf{Swap}(\mathbb R^2\rightarrow V)$, but it is ugly and I wish for a standard or non-standard symbol for this act. I will invent my own if there isn't one.

Thank you in advance for your help!

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    $\begingroup$ Words are nice. Stick with words. $\endgroup$
    – Dan Rust
    Oct 20, 2016 at 22:14

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Most commonly in mathematics, one says in plain English something to the effect of "The proof goes through with $\mathbb{R}^n$ replaced by $V$."

But if you are really looking for a notation, perhaps you might borrow one from lambda calculus. In defining $\beta$-reduction, there is notation for substituting free variables in expressions. I've seen a few notations for "$E$ with the variable $x$ replaced by $E'$":

  • $E[x := E']$ (used on Wikipedia)
  • $E[x \to E']$
  • $E[E'/x]$
  • $[E'/x]E$

I personally find the first notation to be clearest among them.

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  • $\begingroup$ Thank you so much, I truly appreciate your input! I will start using one of these variants after reading the reference! $\endgroup$
    – Andrew Dynneson
    Oct 20, 2016 at 22:41
  • $\begingroup$ P.S. I am trying to write proofs in margins of textbooks, plus I really enjoy the aesthetic beauty of pure notation without English, thank you again. $\endgroup$
    – Andrew Dynneson
    Oct 20, 2016 at 22:46
  • $\begingroup$ I understand your delight in pure symbolic notation, but please be aware that from a logical point of view, you are trying to substitute a variable, (the arbitrary field $\Bbb{F}$) for a constant (the specific field $\Bbb{R}$). This places an obligation on you to check that the proofs you want to generalise don't depend on special properties of $\Bbb{R}$. $\endgroup$
    – Rob Arthan
    Oct 20, 2016 at 23:00
  • $\begingroup$ I agree Rob, I am attempting to clear-out my Linear-Algebra Cobwebs, and extending the standard proofs to $\mathbb F$ finite-dimensional vector spaces (of course checking that they are still legit. and do not reply on any Real numbered properties). Eventually, I want to extend my intuitions to $\Infty $-dimensional Vector Spaces, then Hilbert Space, then possible Bergman Space. Thank you again for your input! $\endgroup$
    – Andrew Dynneson
    Oct 20, 2016 at 23:42

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