Real Analysis | Motivating uniform convergence

[@caldersheagren]

I like to think of uniform convergence as the convergence of a sequence of functions under the supremum norm

[@dbmalesani]

As usual, nice and enlightening video. This may be nitpicking, but in the second example, g_n(x) = ⁿ√x is not differentiable on [0,1] for any n, as it is not differentiable at x = 0.

[@goodplacetostop2973]

15:29

[@sinqobilebandile6558]

These are great. Next video needed expeditiously

[@johnstroughair2816]

Looking forward to the next videos in the series. Can you do one on the implicit function theorem?

[@laurafunk5198]

how can you show that f_n(x)=x/n is not uniformly convergent?

[@FocusHarmonies_1]

If the domain is finite for fn=x/n will it have uniform convergence.

[@duckymomo7935]

Uniform limit theorem: More precisely, let X be a topological space, let Y be a metric space, and let ƒn : X → Y be a sequence of functions converging uniformly to a function ƒ : X → Y. According to the uniform limit theorem, if each of the functions ƒn is continuous, then the limit ƒ must be continuous as well.

[@technoguyx]

that's possibly the best motivation for the concept; uniform convergence is the condition you want for continuity to be preserved under limits, as well as for differentiating and integrating power series "term by term"

[@arvindsrinivasan424]

Gotta love analysis

[@thenewdimension9832]

For understanding your lacture one should be smart enough! Thank-you so much, don't think any video in this topic can be made batter than that.

[@lolatomroflsinnlos]

13:18 forgot a cut?

[@shohbekmahmudov5346]

Hallo ! Help me ! Where can I find suitable books to train for IMC ?

[@valentinamaraenglish8440]

Very nice video.

[@djvalentedochp]

superb!!

[@duckymomo7935]

uniform convergence is a special case of dominated convergence

[@cristianv2850]

I am so confused just looking at this.

[@arturocustodiodiaz]

FIRST !!!!!