Functional Analysis 25 | Hahn–Banach Theorem

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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 Hahn-Banach (extension version)
02:03 Applications

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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)

[cc.quantr]

Functional analysis is also one of my favourites!

[chriswunder]

Great Videos! I'm alwas enjoying the videos after working through theroems in literatur, and it really helps to remember them better as you explain them more playfully.
I'd love you see more aboute sobolev spaces and/or integral transformations.
Anyway great content!

[roozbehr]

It would be beneficial if you make another video of Hahn-Banach Theorem, possibly with more examples maybe and a little explanation (getting into the proof). also how to find all dual norms of a sub space in R^3 maybe. like (x, y, 0) with f(x) = a.x and a=(b, c, 0). ( just an example ). besides, i think Dual spaces worth to be explained more in my opinion. Thanks in advance. i enjoy watching your Series ;)

[vanrltv]

I am a bit confused by all these different norms. I have a few questions if you would be so kind to answer.

1. In part (a), why is ||u'||_u' = 1 ? Where does this 1 come from? The same question for part (c) with sup||x'||_x' .
2. In (c), i am bit confused by notation here. What exactly is a difference between ||x'|| and ||x'||_x'.

I feel like i should know this stuff from previous videos, but i cant seem to connect all the dots.

[wesleyrm]

3:00 -Also, you could have wrote the map u' from U to F as u'(u) = ||u||_X directly.-
-That is, the norm is a linear functional itself.- Of course, it is always positive, unlike your function that maps u=λx to λ||x||_X, which can be negative and seems more useful.

From then Hahn-Banach directly gives us norm preservation in the dual space, so we have x' with ||x'||_X' = ||u'||_U' = sup({|u'(u)|=||u||, for u in U with ||u||=1}) = 1.

The rest of the proof is identical: since x lies in U, x'(x)=u'(x)=||x||_X.

[mathiasbarreto]

I am a little bit confused in part b) of the applications. What exactly is x'(x_1) and x'(x_2) ? Using the property of the map x' in part a), aren't they simply their respective norms? If true won't this mean that different but equidistant vectors (with the same norm) give x'(x_2) - x'(x_1) = 0 although x_1 and x_2 are not the same vectors?

[trondsaue]

In your final example you do not explain why you need a _closed_ subspace U. I believe it is needed to show that the quotient space X/U is a normed vector space.

[kimanthony]

In part (a), $x`(x) := ||x||_X$ seems non-linear function. This seems to go against the definition of dual space. If anyone knows, could you please explain? Thanks.

[trondsaue]

A small question: In part d) you define the quotient space X/U as the equivalence class [x] for x in X. Does the same restriction as two lines above still hold, that is, x is not element of the closed subspace U ? I would say yes, since otherwise the definition becomes strange.

[jaewooknam]

Why do we need a *closed* subspace for (d)? Can you help me to figure it out?

[StratosFair]

Very nice, if you have the time I think some more applications and/or the proof of this theorem would be very helpful