How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional Lp-normed vector space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:32:32Z http://mathoverflow.net/feeds/question/102580 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102580/how-do-you-compute-the-dual-norm-of-an-induced-norm-on-a-subspace-of-a-finite-dim How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional Lp-normed vector space? Mike Battaglia 2012-07-18T20:34:25Z 2012-07-25T06:41:19Z <p>Say you have a finite-dimensional vector space $V$ with an $L^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $L^p$ norm, so the unit sphere in $V_s$ using this induced can be some strange shape.</p> <p>Given $V_s$ and a norm $||·||$ induced this way on it, how can one compute an expression for the dual norm $||·||_*$ on $V_s^*$, the dual space of linear functionals on $V_s$?</p> <p>I understand that this norm must satisfy the relationship $||w||_* = \text{sup }\frac{w(v)}{||v||}$ for $v$ in $V_s$ and $w$ in $V_s^*$, and that this means I need to find the intersection of the unit sphere in $V_s$ with the direction specified by $w$. However, I'm not sure what a good strategy might be to actually find an expression for the dual norm in this way. I thought that some implication of Hahn-Banach might help to pave the way forward, but after some research I still haven't seen anything obvious.</p> <p>I do have a hunch that for the case where the norm on $V$ is $L^1$ or $L^\infty$, and hence where the unit sphere for induced norm on $V_s$ is some sort of polytope, that the unit sphere on $V_s^*$ will be the dual polytope exchanging faces and vertices.</p> http://mathoverflow.net/questions/102580/how-do-you-compute-the-dual-norm-of-an-induced-norm-on-a-subspace-of-a-finite-dim/102584#102584 Answer by Charles Matthews for How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional Lp-normed vector space? Charles Matthews 2012-07-18T20:44:54Z 2012-07-18T20:44:54Z <p>This is a fairly general case of the Legendre transformation, I guess. I don't really see that it should be that much simpler than the general case (but I'm not an expert).</p> http://mathoverflow.net/questions/102580/how-do-you-compute-the-dual-norm-of-an-induced-norm-on-a-subspace-of-a-finite-dim/103075#103075 Answer by Mike Battaglia for How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional Lp-normed vector space? Mike Battaglia 2012-07-25T06:41:19Z 2012-07-25T06:41:19Z <p>An exact solution can be found here using the Hahn-Banach Theorem: <a href="http://math.unl.edu/~s-bbockel1/928/node25.htm" rel="nofollow">http://math.unl.edu/~s-bbockel1/928/node25.htm</a></p> <p>Using this, you can show that $V^ā_S$ is isometrically isomorphic to $V^ā/S$°, where $S°$ is the subspace in Vā for which $s(t)=0$ for $s$ in $S°$ and $t$ in $S$. ā Mike Battaglia Jul 21 at 4:06</p>