Intro Real Analysis, Lec 35: Sup Norm and Metric on C[a,b], Sequence Space, Open & Closed Sets

Introduction to Real Analysis, Lecture 35. Topology Unit, Part 5.

(0:00) Start
(0:36) Graduate level real analysis books.
(2:12) Benefits of this abstract material.
(5:01) Function space: the space of real-valued continuous functions defined on [a,b], with supremum norm (also called "infinity norm" or "uniform norm"), and resulting metric.
(12:25) Idea of proof of the triangle inequality for the sup norm (and thus the triangle inequality is true for the metric as well).
(19:54) "Binary" sequence space "Sigma_2" (the elements are sequences of 0's and 1's) and the metric we will use.
(25:33) The (continuous) shift map on Sigma_2.
(28:52) Fundamental fact: a set is open in a metric space if and only if its (set) complement is closed in that metric space.
(33:25) Outline of proof.
(47:38) The union of any collection of open sets is an open set and the intersection of any finite collection of open sets is an open set. The entire space and the empty set are both open.
(50:57) Relative topology (X is a metric space and A is a subset of X...what does it mean to be open in A?).
(52:49) Introduce application of metric space ideas to studying the behavior of the logistic mappings.

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[gonzalesgrandeyasserchrist]

Muchas gracias me sirvió para mi trabajo del curso, saludos desde el Perú

[greatstuff]

There’s actually also a grand papa Rudin!!! Functional

[maylay]

The same meaning is C(R) and C[a, b]. Why is norm infinity f = C(R), f belong to L^p(R)? Thanks.

[zeeshanahmed]

I have found an mcqs on my test asking that, is f(x)=x on (0, 1) bounded?
I was thinking that on (0, 1) the range of the function is (0, 1) which is bounded so the function should be bounded but relating it to (07:50-07:56) part of video now I am thinking that it should be unbounded because on (0, 1) we will get no maximum value of the function.
So what should be the right answer bounded or unbounded?

[advancedappliedandpuremath]

Sir why we can't take infimum instead of supremum