Friendship doesn't necessarily mean you are trying to get into his/her underwear!. If you mean is it OK to remain friends outside the framework of the flying lesson, then I see no problem. I have made a few friends through instructing and we still fly together as fellow pilots.
Not quite sure what you mean by 'inappropriate'. If we are talking about it affecting the necessarily commercial nature of a flying lesson then that is up to you to decide. If your flying club thinks your friendship is affecting their business (e.g. charging 0.7 for a 1.0 lesson) then you are on a sticky wicket.
However, if you mean what everybody else thought, then it is like any other work romance: don't let it affect your professional outlook and keep the romantic bit for outside the a/c. Make sure you keep your detached judgement of ability/progress. If you're trying it on on the offchance: watch out for harrassment claims; that is unprofessional and maybe illegal!!
Strange question, really..
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"Take-off is optional, Landing is mandatory"