The letter ⟨u⟩ is used in text messaging and internet and other written slang to denote 'you', by virtue of both being pronounced /juː/.
When and by whom was the letter "u" first used as an abbreviation for the word "you" in writing?
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According to the Oxford English Dictionary, it has been used in IOU since 1795:
An I.O.U. is admissible evidence of a debt without a stamp.
Espinasse's Rep. (margin note)
As for standalone examples, the first one given by the OED is from 1840:
There was no one but the plaintiff to whom the ‘U’ in the document [an IOU] could be applied. Manning and Granger's Reports
The next oldest example given is from 1862, in a letter from Thomas Hardy to his sister:
I wish you wd tell me how u.r. when u. write.
After that, the OED lists a source which mentions it in an advertising context:
This is another device which appears to be growing in popularity. Where possible a capital letter is often substituted for writing out of a syllable.
Call Tel-U-Where for our nearest distributor... Fits-U Eyeglasses. U All Kno After Dinner Mints. Uneeda Biscuit.