At 35:25, you mark the proof of the open mapping theorem as complete. I'm not sure that it is. Maybe it's my own ignorance. I've never taken any analysis or topology courses. I've only watched some lectures on RUclips, including a few from your course on real analysis. It seems to me that you've shown that T(U) contains an open ball, namely B(b2, epsilon*delta). But I don't see how this implies that T(U) is open (I'm assuming you mean open with respect to the topology on B2 induced by the norm). This is especially confusing for me since, in Baire's theorem, it was shown that at least one of those *closed* subsets (given by the conditions in the theorem) contains an open ball. Am I missing something?
If I'm not mistaken, what he has shown is that for any point b_2\in T(U), there is an open ball containing b_2 and contained in T(U). This suffices to show that T(U) is open.
@@yoshka23 mathematics is an art more than people realize, this shouldn't be thought of as used in life, but something done for its own sake for fufillment
If you have a smartphone or laptop, the probability that you are benefitting in a real way, right now, from the achievements of functional analysis is practically 1. Iirc, the motivation behind functional analysis is to generalize the solution methods of systems of linear differential equations. Pretty much all advanced simulation software used in engineering and physics implement an optimized algorithm for arriving at solutions to a relevant system of differential equations, with the boundary conditions defined by the user. It's highly likely that some theorems from functional analysis are often used to optimize those algorithms. (I'm qualifying this statement only because I've never written that kind of software myself, though I have used it for electrical engineering). A notable example would be the simulation software used to design antennas. This is especially true for the planar antennas frequently found in consumer electronics. The antennas in your laptop or smartphone (for wifi, Bluetooth, cellular data, GPS, etc.) almost certainly were designed using that kind of software, which itself almost certainly used results from this subfield of mathematics to speed up their simulation times. Faster simulation times means faster and easier design and development, which means less engineering overhead, which means cheaper and better working devices.
![Functional Analysis 26 | Open Mapping Theorem [dark version]](/img/1.gif)
"So let's continue with the 'big-name theorems' or the 'theorems with big names' or the 'big theorems with names'", absolutely hilarious!
35:37 End of proof to open mapping theorem
57:21 End of proof to closed graph theorem
Thanks mit for providing high quality education
Open mapping theorem is the most used result from functional analysis
At 35:25, you mark the proof of the open mapping theorem as complete. I'm not sure that it is. Maybe it's my own ignorance. I've never taken any analysis or topology courses. I've only watched some lectures on RUclips, including a few from your course on real analysis.
It seems to me that you've shown that T(U) contains an open ball, namely B(b2, epsilon*delta). But I don't see how this implies that T(U) is open (I'm assuming you mean open with respect to the topology on B2 induced by the norm). This is especially confusing for me since, in Baire's theorem, it was shown that at least one of those *closed* subsets (given by the conditions in the theorem) contains an open ball. Am I missing something?
If I'm not mistaken, what he has shown is that for any point b_2\in T(U), there is an open ball containing b_2 and contained in T(U). This suffices to show that T(U) is open.
@@ceyhunelmacioglu9819this argument is correct. It is a fundamental property of open sets.
Thanks for asking and thanks for the answer guys! Helped a lot.
I remember when the 4 color was a lived controversy.
45:23
Nice 45:23
When would any of this be used in life?
free to go elsewhere...
@@pspspspssspspps You couldn’t even answer my question. 😂
Build (and prove) robust algorithms that solve engineering problems from buildings to your phone.
@@yoshka23 mathematics is an art more than people realize, this shouldn't be thought of as used in life, but something done for its own sake for fufillment
If you have a smartphone or laptop, the probability that you are benefitting in a real way, right now, from the achievements of functional analysis is practically 1.
Iirc, the motivation behind functional analysis is to generalize the solution methods of systems of linear differential equations. Pretty much all advanced simulation software used in engineering and physics implement an optimized algorithm for arriving at solutions to a relevant system of differential equations, with the boundary conditions defined by the user. It's highly likely that some theorems from functional analysis are often used to optimize those algorithms. (I'm qualifying this statement only because I've never written that kind of software myself, though I have used it for electrical engineering).
A notable example would be the simulation software used to design antennas. This is especially true for the planar antennas frequently found in consumer electronics. The antennas in your laptop or smartphone (for wifi, Bluetooth, cellular data, GPS, etc.) almost certainly were designed using that kind of software, which itself almost certainly used results from this subfield of mathematics to speed up their simulation times. Faster simulation times means faster and easier design and development, which means less engineering overhead, which means cheaper and better working devices.