My three screencasts on How and why we do mathematical proofs are available below (you can also find them on YouTube).
See also the University of Nottingham’s Open Courseware U-Now module “How and why we do proofs”, available at
http://unow.nottingham.ac.uk/resources/resource.aspx?hid=9ceaa739-b7a0-3c49-fb87-52b6dcb47c5e
Explaining mathematics 
Hey Joel, a nice introduction. Are you going to give your students any hints about Gödel’s Theorem?
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Hi,
i am a Civil Engineer who is enrolled at Phd program in Spain. I want to learn the language of pue mathematics, and later do research on Numerical/Functional Analysis. What path should i follow. I would apreciate any suggestion you can give me. Regards,
VMHP
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It does take time to develop the language and thinking of pure mathematics, so it won’t be quick!
Of course there are lots of good books you can look at.
Some of the materials I have made available might help a bit. (Of course there is also plenty on YouTube/iTunes from Berkeley/MIT/etc.).
There are my videos on How and why we do mathematical proofs, and on Definitions, Proofs, and Examples. There is my complete set of videos on second-year Mathematical Analysis. There is my complete set of annotated slides and audio recordings from my old 3rd-year module on Measure and Integration. There is the complete set of videos from my 4th-year module on Functional Analysis. That all comes to a lot of hours!
Of course it may be that you could take a more “Applied Functional Analysis” approach, which might not require you to dot the i’s and cross the t’s so much.
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