tag:blogger.com,1999:blog-391332808443807350.post599921336666419125..comments2020-11-20T03:03:44.658-05:00Comments on gtMath: Convergence of Sequences of Functionsgtmathhttp://www.blogger.com/profile/17592451309514283248noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-391332808443807350.post-51086184913976603682016-09-05T08:46:06.302-04:002016-09-05T08:46:06.302-04:00Thanks, Charles- great question! I am actually pla...Thanks, Charles- great question! I am actually planning on including this in an upcoming post, so stay tuned. You can enter your email in the "Follow by Email" box on the right to be notified of new posts without having to keep checking the homepage.<br /><br />By the way, there is also a separate Reader Request page I set up which you can use to post requests for new topics: http://www.gtmath.com/p/reader-requests.htmlgtmathhttps://www.blogger.com/profile/17592451309514283248noreply@blogger.comtag:blogger.com,1999:blog-391332808443807350.post-66207600812963542682016-09-02T13:53:58.110-04:002016-09-02T13:53:58.110-04:00Troderman,
I'm enjoying the blog! This post g...Troderman,<br /><br />I'm enjoying the blog! This post got me thinking about specific examples of metric spaces generated by norms and interesting properties that can arise. It might be out in the weeds but I never figured out the proof relating the $l^p(x)$ to $l^q(x)$ given in the theorem below. The case becomes trivial when $p=q=2$, since you get the added structure of a Hilbert Space. It could be an interesting question to pursue, there's a lot going on with it! (And i'd love to see a solution explained.)<br /><br />Let $l^p$ be a subspace of the space of all sequences of scalars such that $\sum_{n} |x|^p < \infty$ equipped with the usual norm $\|x\|_p = (\sum_n |x_n|^p)^\frac{1}{p}$ <br /><br /><b>Theorem:</b> The duel space of $l^p(x)$ for $1 < p < ∞$ has an isomorphism with $l^q(x)$, where $q$ is such that $\frac{1}{p}$ +$\frac{1}{q} = 1$Charles Stephenshttps://www.blogger.com/profile/16639389289621574266noreply@blogger.com