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.
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.
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"
I like to think of uniform convergence as the convergence of a sequence of functions under the supremum norm
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.
how can you show that f_n(x)=x/n is not uniformly convergent?
15:29
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.
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"
I am so confused just looking at this.
Gotta love analysis
FIRST !!!!!
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.
Hallo !
Help me ! Where can I find suitable books to train for IMC ?
Looking forward to the next videos in the series. Can you do one on the implicit function theorem?
uniform convergence is a special case of dominated convergence
If the domain is finite for fn=x/n will it have uniform convergence.
These are great. Next video needed expeditiously
13:18 forgot a cut?
Very nice video.
superb!!