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

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

I was thinking Swap(R2→V)\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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Words are nice. Stick with words.

– Dan Rust

Oct 20 at 22:14

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1 Answer

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

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

E[x:=E′]E[x := E’] (used on Wikipedia)

E[x→E′]E[x \to E’]

E[E′/x]E[E’/x]

[E′/x]E[E’/x]E

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

Thank you so much, I truly appreciate your input! I will start using one of these variants after reading the reference!

– Andrew Dynneson

Oct 20 at 22:41

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.

– Andrew Dynneson

Oct 20 at 22:46

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 F\Bbb{F}) for a constant (the specific field R\Bbb{R}). This places an obligation on you to check that the proofs you want to generalise don’t depend on special properties of R\Bbb{R}.

– Rob Arthan

Oct 20 at 23:00

I agree Rob, I am attempting to clear-out my Linear-Algebra Cobwebs, and extending the standard proofs to F\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\Infty -dimensional Vector Spaces, then Hilbert Space, then possible Bergman Space. Thank you again for your input!

– Andrew Dynneson

Oct 20 at 23:42