This is mostly aimed after the 1261 recapture of Constantinople. Im interested to know if the Byzantine Emperors, people, and officials viewed their fall, and how they reacted to it.
Edit: Hello all, I didn't expect this to get as many upvotes as it did, so I thank you all a ton. However, the question, I believe has been misinterpreted. I am not asking how it fell, I am asking how the Byzantine people, especially the actual Emperors viewed it, and if they did take note of their own decline, if there is any evidence of a strong effort to reverse the fates. Again, thanks, and hopefully we can all get an answer!
I wrote this reply in response to a comment that got deleted asking about Byzantine aristocrats fleeing. I thought I'd post it anyway.
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I don't know much about aristocrats fleeing (I'm sure they did), but there are two important points.
First the Eastern Roman Empire had a policy of using royal marriage as a diplomatic tool. Many a young Byzantine/Greek princesses were married off to other European royalty as a form of diplomacy.
For example Moscow's claims to being the Third Rome was partly based on Ivan III marrying the niece of the last Byzantine Emperor, Zoe/Sophia Palaiologina.
However, more importantly is number of Greek scholars who fled to Italy (especially after the Ottoman conquest) but also before it. These scholars made significant contributions to the Italian Renaissance.
Some argue that it was the wave of Greek scholars coming to Italy after the fall of Constantinople which was the catalyst that triggered the Italian Renaissance.
See this wikipedia article on Greek scholars in the Renaissance