Скачать или смотреть Lecture 2: Linear spaces

Lecture 2 lays down the basics for linear spaces. We start with the characteristics regarding elements and operations that make a set a linear space. Then we examine concepts of fundamental importance like subspaces, linear combinations, and linear independence. Due to the significance in applications, most attention is paid to convexity and convergence. To understand the latter, we need to introduce norms and metrics, and normed linear spaces. Even though our textbook doesn’t cover metrics and metric spaces, we refer to Stokey and Lucas (1989) to see that norms can be used as metrics. We also deal with the Hölder inequality for infinite sequences and measurable real functions that proves to be useful in certain types of maximum or minimum problems.

Title page:(00:00)
Linear spaces and their operations:(00:10)
Subspaces and linear combinations:(07:45)
Problem 3 from page 43:(14:10)
Translation of sets, and linear varieties:(27:58)
Convexity, convex cones, and linear independence:(30:57)
Problem 4 from page 43:(35:39)
Normed linear spaces:(55:56)
Normed linear vector spaces:(59:35)
Open, closed sets, and convergence:(65:33)
The lp and Lp spaces, and the Hölder inequality:(72:04)
The Hölder inequality for functions:(75:55)
Solving a maximum problem with the Hölder inequality:(77:01)
Solving a minimum problem with the Hölder inequality:(96:17)