This video explains the definition of a vector space and provides examples of vector spaces.
From playlist Vector Spaces
In this video, I discuss the notion of sequential compactness, which is an important concept used in topology and analogy. I also explain the similarities and differences between sequential compactness and covering compactness. Compactness: https://youtu.be/xiWizwjpt8o Bolzano-Weierstrass
From playlist Topology
MAST30026 Lecture 9: Compactness II (Part 1)
I defined open covers and compactness for topological spaces, and proved that if a metric space is sequentially compact then the associated topological space is compact. Lecture notes: http://therisingsea.org/notes/mast30026/lecture9.pdf The class webpage: http://therisingsea.org/post/mas
From playlist MAST30026 Metric and Hilbert spaces
This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.
From playlist Proofs
Every Compact Set in n space is Bounded
Every Compact Set in n space is Bounded If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Thank you:)
From playlist Advanced Calculus
We have already looked at the column view of a matrix. In this video lecture I want to expand on this topic to show you that each matrix has a column space. If a matrix is part of a linear system then a linear combination of the columns creates a column space. The vector created by the
From playlist Introducing linear algebra
A WEIRD VECTOR SPACE: Building a Vector Space with Symmetry | Nathan Dalaklis
We'll spend time in this video on a weird vector space that can be built by developing the ideas around symmetry. In the process of building a vector space with symmetry at its core, we'll go through a ton of different ideas across a handful of mathematical fields. Naturally, we will start
From playlist The New CHALKboard
Metric space definition and examples. Welcome to the beautiful world of topology and analysis! In this video, I present the important concept of a metric space, and give 10 examples. The idea of a metric space is to generalize the concept of absolute values and distances to sets more gener
From playlist Topology
What is a Vector Space? Definition of a Vector space.
From playlist Linear Algebra
Metric Spaces - Lectures 19 & 20: Oxford Mathematics 2nd Year Student Lecture
For the first time we are making a full Oxford Mathematics Undergraduate lecture course available. Ben Green's 2nd Year Metric Spaces course is the first half of the Metric Spaces and Complex Analysis course. This is the 10th of 11 videos. The course is about the notion of distance. You m
From playlist Oxford Mathematics Student Lectures - Metric Spaces
Metric Spaces - Lectures 21, 22 & 23: Oxford Mathematics 2nd Year Student Lecture
For the first time we are making a full Oxford Mathematics Undergraduate lecture course available. Ben Green's 2nd Year Metric Spaces course is the first half of the Metric Spaces and Complex Analysis course. This is the 11th of 11 videos. The course is about the notion of distance. You m
From playlist Oxford Mathematics Student Lectures - Metric Spaces
Metric Spaces - Lectures 17 & 18: Oxford Mathematics 2nd Year Student Lecture
For the first time we are making a full Oxford Mathematics Undergraduate lecture course available. Ben Green's 2nd Year Metric Spaces course is the first half of the Metric Spaces and Complex Analysis course. This is the 9th of 11 videos. The course is about the notion of distance. You ma
From playlist Oxford Mathematics Student Lectures - Metric Spaces
MAST30026 Lecture 8: Compactness I
This is the first of several lectures on compactness. I recalled the proof of the Bolzano-Weierstrass theorem, defined sequential compactness in metric spaces and the characterisation of continuity of functions in terms of limits, and proved that the image of a compact set is compact. Lec
From playlist MAST30026 Metric and Hilbert spaces
MAST30026 Lecture 9: Compactness II (Part 2)
This lecture contains the proof that compact implies sequentially compact, that continuous images of compact sets are compact, and I finished by observing this gives us the Extreme Value Theorem for compact topological spaces. Lecture notes: http://therisingsea.org/notes/mast30026/lecture
From playlist MAST30026 Metric and Hilbert spaces
Math 131 Fall 2018 103118 Introduction to lim sup and lim inf
Recall: Cauchy sequence; complete metric space. Theorem: Euclidean space is complete. Monotonic sequences. Monotonic Sequence Theorem. Definition of limit superior, limit inferior. Examples. Statement of characterization of lim sup. Comment: if lim sup equals lim sup, then sequence
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)
Sequential Spectra- Part 5: Spectrification
The second part of the Omega spectra section on nLab. Credits: nLab: https://ncatlab.org/nlab/show/Introdu... Animation library: https://github.com/3b1b/manim Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • YouTube Track Link: https://bi
From playlist Sequential Spectra
MAST30026 Lecture 10: Compactness III
In this lecture I explained how the compactness property is inherited by spaces constructed according to our standard tools (products, disjoint unions, quotients). We proved Heine-Borel and that finite CW-complexes are compact. Lecture notes: http://therisingsea.org/notes/mast30026/lectur
From playlist MAST30026 Metric and Hilbert spaces
Math 131 Fall 2018 101018 Continuity and Compactness
Definition: bounded function. Continuous image of compact set is compact. Continuous image in Euclidean space of compact set is bounded. Extreme Value Theorem. Continuous bijection on compact set has continuous inverse. Definition of uniform continuity. Continuous on compact set impl
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)