Functional Analysis 14 | Example Operator Norm

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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
00:15 Example
01:27 First estimate
03:30 Second estimate

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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)

[yuanxiwu]

Very nice course. You really make it easier for people majoring in engineering to understand functional analysis! Have supported on steady~

[zazinjozaza]

Functional analysis is really fun! Thank you for making these videos!

[RangQuid]

All your explanation and demonstration are extremely rigorous, I love it 😊😊!

[alirezaghadami]

Wow, these videos are amazing and very helpful, keep it up 👍

[muhammedali]

Could you please explain why at 1:46 we are able to take the supremum of f to be 1 without loss of generality? Aren't we looking for the supremum over all f in X?

[edgarlangwald]

But h is not continous, so Th shouldn’t be defined, right? Wouldn‘t you habe to argue with a sequence of continous functions that converges to h?

[xwyl]

This time construction of h(t) is a bit easier to understand: to make h(t) as big as possible to maximize the integral.

[qiaohuizhou]

Hi, I don't quite get why at 1:55 we can simply choose f such that the ||f|| =1 without changing the equality of the sup? Could you please explain that a bit?

[qingninghuo]

At 0:32, Can u please clarify how the supreme norm of a continuous function is defined? Is this defined in one of the earlier video of this series? Thank you.

[張毓倫-lc]

Teacher, why is the absolute value of f(x) less than the norm of f at 2:39 in the video?

[abhayrao]

Wonderful and informative video.
I had a query. Can you kindly explain how, while defining the operator norm of ||Tg||=norm of ouput by norm of input (1:27 onwards) is shown as absolute of output by norm of input. i.e shdnt it be ||Tg|| = { ||Tg(f)|| / ||f|| } but the video shows.... ||Tg|| = { |Tg(f)| / ||f|| }. Can you kindly clarify and correct me where I am going wrong? Thank you soo much

[ahmedamr]

Great video!
Isn't Tg(f) an inner product over X?

[stellamn]

i don't see how you can assume there is no loss in generality when choosing the function h to be the complex conjugate of g. how do you recreate any other function from h and your lower bound remains true?

[rohankapoor]

why did you include the condition of g having no zeroes?