Functional Analysis 12 | Continuity [dark version]

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This is my video series about Functional Analysis where we start with metric spaces, talk about operators and spectral theory, and end with the famous Spectral Theorem. I hope that it will help everyone who wants to learn about it.
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00:00 Introduction
00:20 Definition - continuity
09:56 Orthogonal complement is closed
#FunctionalAnalysis
#Mathematics
#LearnMath
#calculus
I hope that this helps students, pupils and others. Have fun!
(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[@elsa1569]

This video has so many great examples, using fundamental results!

[@psolien]

Slight nit pick for (c) we can't take lim f() as we don't know if it exists. We use the inequalities to form an upper bound for the absolute value of the difference (standard metric in R) instead

[@brightsideofmaths]

One can write it down with limsup and liminf but I wanted to keep it short :)

[@natepolidoro4565]

6:38 Don't we have to know that the metric is continuous in one of it's entries to say this?

[@brightsideofmaths]

Yes: every metric is continuous simply by definition of continuity in the metric space.