Question

In: Advanced Math

definitions of weak topology and weak star topology with as little jargon as possible.

Expert Solution

Let be a non empty set and be a family of topological spaces. Let . Then we define weak topology on in such a way that each is continuous with respect to in the following way:

Since we expect each to be continuous with respect to , if , then . For to be a topology, it should be closed under arbitrary unions. Let , then the topology generated by is the required topology, i.e., is the basis of the weak topology generated by .

Let be a normed space and be its dual space. For each , the functional , is a semi-norm on . The topology induced by the family of seminorms is the weak topology on . Thus, weak star topology is nothing but the weak topology generated on the dual space by the family of functions which will be continuous in this topology.

The aim to define this kind of topologies is to simplify our study of the space of continuous functions because every function on the topological space (with weak topology) is continuous.

Colby Messinger answered 2 years ago

While there are several topologies and technologies used in WANs, the star topology with Ethernet is...

While there are several topologies and technologies used in WANs, the star topology with Ethernet is the dominant topology/technology pair in LANs. Consider why this is so. Think about properties such as management, reliability, security, costs, and so forth. Instructions: Write a post of at least two paragraphs discussing why you think the tandem of star topology/Ethernet is dominant in LAN designs. Include some of the advantages that other topologies/technologies have over this tandem and try to explain why these...

Design and configure the Star Topology in Packet Tracer with following requirements. • 4 PCs •...

Design and configure the Star Topology in Packet Tracer with following requirements. • 4 PCs • 1 SWITCH • Connection Cable

For a relatively weak gravitational field, such as that of a planet or an ordinary star,...

For a relatively weak gravitational field, such as that of a planet or an ordinary star, the following formula tells us the fractional amount of gravitational time dilation at a distance r from the center of an object of mass Mobject: 1c2×GMobjectr (G=6.67×10?11m3/(kg×s2); c=3×108m/s.) For example, while 1 hour passes in deep space far from the object, the amount of time that passes at a distance r is 1 hour minus 1 hour multiplied by the factor above. (This formula...

b. Caitlin is a little concerned that the effect of relationship quality will be weak, and...

b. Caitlin is a little concerned that the effect of relationship quality will be weak, and she wants to be sure to detect it. What can she do to make sure her math problem’s dependent variable is a sensitive measure of problem solving? c. What can she do to make sure her math problem’s dependent variable is a construct valid measure of problem solving? d. What else can Caitlin do to make sure she has the best study possible to...

Prove that the discrete topology on X is the same as the metric topology induced by...

Prove that the discrete topology on X is the same as the metric topology induced by the discrete metric. Where metric topology is defined as: If (X,d) is a metric space, then consider the collection T of all open subsets of X. Then (X,T) is topological space. This topology is called the metric topology on X induced by d.

a) the possible definitions and determinants of reputational risk and its insurance; b) the legal and...

a) the possible definitions and determinants of reputational risk and its insurance; b) the legal and regulatory frameworks of reputational risk and its insurance; c) which types of modern organisations would benefit most from reputational risk insurance; d) how reputational risk can be quantified for insurance purposes; e) the technical terms and conditions of reputational risk insurance; You may support your discussions and explanations with examples where applicable. However, calculations are optional.

As far as possible, assign the following terms to the definitions and descriptions below. To do...

As far as possible, assign the following terms to the definitions and descriptions below. To do this, enter the corresponding letter in the table to the left of the term. If no definition matches the term, please enter "-". Definitions and descriptions: A. Term for a quantity expressed in terms of a point in time (e.g. capital stock). B. Increases net welfare in the presence of negative external effects. C. Selling price minus costs of a good. D. ...

In Star Wars Episode 4: A New Hope, Princess Leia says Luke is "a little short...

In Star Wars Episode 4: A New Hope, Princess Leia says Luke is "a little short to be a storm trooper." To test this we went to New Orleans Comic-Con and took the heights, in inches, of Luke cosplayers and Storm Trooper cosplayers, the data is in the table below. Lukes Storm Troopers x L ¯ = 57 x S ¯ = 60 n L = 10 n S = 15 s L = 3 s S = 2 Use...

I need these terms and definitions answered, as much as possible is appreciated. Xenophobia: intense or...

I need these terms and definitions answered, as much as possible is appreciated. Xenophobia: intense or irrational dislike or fear of people from other countries. big data: extremely large data sets that may be analyzed computationally data measurements: bit , yottabyte Planned obsolescence : designing and producing products in order for them to be used up (obsolete) within a specific time period. Perceived obsolescence: an object may continue to be functional, but it is no longer perceived to be stylish...

Topology (a) Prove that the interval [0,1] with the subspace topology is connected from basic principles....

Topology (a) Prove that the interval [0,1] with the subspace topology is connected from basic principles. (b) Prove that the interval [0,1] with the subspace topology is compact from basic principles.