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