{"id":543,"date":"2012-10-16T05:00:31","date_gmt":"2012-10-16T12:00:31","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=543"},"modified":"2012-10-15T21:18:57","modified_gmt":"2012-10-16T04:18:57","slug":"bounded-but-not-totally-bounded-redux","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2012\/10\/bounded-but-not-totally-bounded-redux\/","title":{"rendered":"Bounded but Not Totally Bounded, Redux"},"content":{"rendered":"<p><em>Theorem<\/em>. An open set in real sequence space under the \u2113<sup>\u221e<\/sup> norm is not totally bounded.<\/p>\n<p><em>Proof<\/em>. Consider an open set <em>U<\/em> containing a point <em>p<\/em>. Suppose by way of contradiction that <em>U<\/em> is totally bounded. Then for every \u03b5 &gt; 0, there exists a finite \u03b5-net for <em>U<\/em>. Fix \u03b5, and let <em>m<\/em> be the number of points in our \u03b5-net, which net we'll denote {<em>S<\/em><sub><em>i<\/em><\/sub>}<sub><em>i<\/em>\u2208{1, ..., <em>m<\/em>}<\/sub>. We're going to construct a very special point <em>y<\/em>, which does not live in <em>U<\/em>. For all <em>i<\/em> \u2208 {1, ..., <em>m<\/em>}, we can choose the <em>i<\/em>th component <em>y<\/em><sub><em>i<\/em><\/sub> such that the absolute value of its difference from the <em>i<\/em>th component of the <em>i<\/em>th point in the net is strictly greater than \u03b5 (that is, |<em>y<sub>i<\/sub><\/em> &ndash; <em>S<sub>i,i<\/sub><\/em>| &gt; \u03b5) but also so that the absolute value of its difference from the <em>i<\/em>th component of <em>p<\/em> is less than or equal to \u03b5 (that is, |<em>y<sub>i<\/sub><\/em> \u2013 <em>p<sub>i<\/sub><\/em>| \u2264 \u03b5). Then for <em>j<\/em> &gt; <em>m<\/em>, set <em>y<sub>j<\/sub><\/em> = <em>p<sub>j<\/sub><\/em>. Then |<em>y<\/em> \u2013 <em>p<\/em>| \u2264 \u03b5, but that means there are points arbitrarily close to <em>p<\/em> which are not in <em>U<\/em>, which is an absurd thing to happen to a point in an open set! But that's what I've been trying to tell you this entire time.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Theorem. An open set in real sequence space under the \u2113\u221e norm is not totally bounded. Proof. Consider an open set U containing a point p. Suppose by way of contradiction that U is totally bounded. Then for every \u03b5 &hellip; <a href=\"http:\/\/zackmdavis.net\/blog\/2012\/10\/bounded-but-not-totally-bounded-redux\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[33],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/543"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=543"}],"version-history":[{"count":11,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/543\/revisions"}],"predecessor-version":[{"id":554,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/543\/revisions\/554"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=543"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=543"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=543"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}