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