{"id":1480,"date":"2017-11-16T02:15:35","date_gmt":"2017-11-16T07:15:35","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1480"},"modified":"2017-11-26T12:35:16","modified_gmt":"2017-11-26T17:35:16","slug":"daily-definitions-1114","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/11\/16\/daily-definitions-1114\/","title":{"rendered":"Daily Definitions 11\/14"},"content":{"rendered":"

In this definitions blog, I will explore the supremum norm in greater detail. First, recall the definition of the supremum norm: If \"f,g:A\rightarrow\mathbb{R}\", then we set the supremum norm \"\mid\mid. for the “uniform size” of \"f\" and \"\mid\mid is the “uniform distance” between \"f\" and \"g\".<\/p>\n

Notice that the word “uniform” comes up in this definition. Recall previous concepts of both uniform continuity and uniform convergence. With both of these ideas, the “uniformity” is characterized by a lack of dependence on \"x\". Well, this same idea is also true for the supremum norm.<\/p>\n

Extending upon this idea, it is reasonable to entertain the idea of a connection between uniform convergence and the “uniform distance” between functions. Recall the proposition made in class, that \"f_n\rightarrow uniformly on \"A\" iff \"\mid\mid.<\/p>\n

We proved this statement to be true. However, consider the statement: If \"f_n\rightarrow pointwise on \"A\", then \"\mid\mid.<\/p>\n

This statement is not true. Consider the sequence of functions \"f_n=\frac{x^2+nx}{n}\". \"f_n\rightarrow point wise, where \"f(x)=x\". However, notice,
\n$\"\sup\mid$<\/p>\n

But \"\mid\frac{x^2}{n}\mid\" is unbounded, so this supremum does not exists by the AoC. Thus, we see uniform convergence is required.<\/p>\n

Now, it follows that \"f_n\" also lack uniform convergence by our first proposition. With this new understanding, we can now use the supremum norm as a nice tool to prove absence of uniform convergence in the future.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this definitions blog, I will explore the supremum norm in greater detail. First, recall the definition of the supremum norm: If , then we set the supremum norm . for the “uniform size” of and is the “uniform distance” between and . Notice that the word “uniform” comes up in this definition. Recall previous […]<\/p>\n","protected":false},"author":3526,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[58823],"tags":[],"class_list":["post-1480","post","type-post","status-publish","format-standard","hentry","category-definitions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-nS","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1480","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/3526"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1480"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1480\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1480"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1480"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1480"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}