Where to start learning economics as a mathematician? I study pure mathematics at university and I'm looking to get into economics, with pretty much no background knowledge about it.

Are there good sources to learn from with a decent mathematical background in mind, or should I just start learning from "normal" sources, like watching video lectures from introductory courses, which don't assume much mathematical knowledge? Preferably, I'd like to embrace economics without those limitation.
McCloskey's Applied Theory of Price is somewhat legendary as a model of clarity. It is, sadly, no longer in print. However, a PDF is available here: http://www.deirdremccloskey.com/docs/price.pdf. The level of mathematical sophistication in this book (like in most good economics!) is not that high, but the book gives a deeper look at the important concepts than is to be found in most extant introductory undergraduate texts.

If you really want a more mathematically formalised treatment then you are going to have to turn to graduate-level econ books. For microeconomics, the classic text of choice is Microeconomic Theory by Mas-Colell et al.

If you do go down the more formal route, I would urge you not to give up on simpler treatments altogether; they're great for building intuition. More than a decade of experience in economics has taught me that good intuition built through simple models is often far more valuable than a less intuitive understanding of a more sophisticated model.