When a Banach space is a Hilbert space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:20:20Z http://mathoverflow.net/feeds/question/11192 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11192/when-a-banach-space-is-a-hilbert-space When a Banach space is a Hilbert space? Teiko Heinosaari 2010-01-08T21:46:10Z 2011-12-07T11:03:20Z <p>Let $\mathcal{X}$ be a Banach space. It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds.</p> <p>Question: Are there other (simple) characterizations for a Banach space to be a Hilbert space?</p> http://mathoverflow.net/questions/11192/when-a-banach-space-is-a-hilbert-space/11194#11194 Answer by Steve Huntsman for When a Banach space is a Hilbert space? Steve Huntsman 2010-01-08T22:05:02Z 2010-01-08T22:05:02Z <p>From <a href="http://www.springerlink.com/content/731n45405715p2m3/" rel="nofollow">http://www.springerlink.com/content/731n45405715p2m3/</a></p> <blockquote> <p>a real Banach space (X, ‖ · ‖) is a Hilbert space if and only if for any three points A,B,C of this space not belonging to a line there are three altitudes in the triangle ABC intersecting at one point.</p> </blockquote> <p>Many other references show when Googling</p> <blockquote> <p>"is a hilbert space if" banach</p> </blockquote> http://mathoverflow.net/questions/11192/when-a-banach-space-is-a-hilbert-space/11195#11195 Answer by Jonas Meyer for When a Banach space is a Hilbert space? Jonas Meyer 2010-01-08T22:16:56Z 2010-01-08T22:16:56Z <p>Yes, there are many (simple) characterizations of when a normed space is an inner product space. Here are two book references, <a href="http://books.google.com/books?id=nBWsDtVfsbsC&amp;lpg=PP1&amp;client=firefox-a&amp;pg=PA110#v=onepage&amp;q=&amp;f=false" rel="nofollow">one</a> with Google preview, the <a href="http://books.google.com/books?id=VVmqAAAAIAAJ&amp;client=firefox-a&amp;source=gbs%5Fnavlinks%5Fs" rel="nofollow">other</a> you can hopefully get at your library.</p> http://mathoverflow.net/questions/11192/when-a-banach-space-is-a-hilbert-space/19912#19912 Answer by Ady for When a Banach space is a Hilbert space? Ady 2010-03-31T03:07:08Z 2010-03-31T03:07:08Z <p>Just two isometric/isomorphic characterizations:</p> <p>A Banach space $X$ is [isometric to] a Hilbert space if and only if there exists a Banach space $Y$ and a symmetric bilinear mapping $f:X\times X\rightarrow Y$ satisfying </p> <p>$||f(x,z)||$ $=$ $||x||\cdot||z|$| for all $x,z$ $\in$ $X$. </p> <p>[J. Becerra Guerrero &amp; A. Rodriguez-Palacios]</p> <p>A Banach space is [isomorphic to] a Hilbert space iff it is uniformly homeomorphic to a Hilbert space. [Per Enflo]</p> http://mathoverflow.net/questions/11192/when-a-banach-space-is-a-hilbert-space/20754#20754 Answer by BigBill for When a Banach space is a Hilbert space? BigBill 2010-04-08T16:27:16Z 2010-04-08T16:27:16Z <p>Bessaga and Pelczynski wrote a survey on Banach spaces. The chapter 4 is devoted to this question. </p> <p><a href="http://matwbn.icm.edu.pl/ksiazki/or/or2/or214.pdf" rel="nofollow">http://matwbn.icm.edu.pl/ksiazki/or/or2/or214.pdf</a></p> http://mathoverflow.net/questions/11192/when-a-banach-space-is-a-hilbert-space/57089#57089 Answer by Garrisi Daniele for When a Banach space is a Hilbert space? Garrisi Daniele 2011-03-02T08:30:14Z 2011-03-06T10:38:05Z <p>More characterisations are in the book of Haim Brezis (Analyse fonctionnelle), at the appendix of Chapter 5. I will copy two of these below, toghether with the references:</p> <ol> <li>If $\dim(E)\geq 2$ and every subspace $X\subset E$ of dimension $2$ is the image of a bounded projector $P$ such that $\|P\| = 1$, then $E$ is isometric to a Hilbert space (Kakutani, <em>Japanese Journal of Mathematics</em>, 1939);</li> <li>if $\dim(E)\geq 3$ and the map $T$, defined as the identity on the unit ball and as $u/\|u\|$ when $\|u\|\geq 1$, is lipschitzian with constant $1$, then $E$ is isometric to a Hilbert space (de Figueiredo; Karlovitz, <em>Bulletin of the American Mathematical Society</em>, 1967).</li> </ol> <p>Also, if $E$ is isomorphic to all its infinite-dimensional subspaces, then it is isomorphic to a separable Hilbert space (Gowers, <em>Annals of Mathematics</em>, 2002).</p> http://mathoverflow.net/questions/11192/when-a-banach-space-is-a-hilbert-space/77438#77438 Answer by Valerio Capraro for When a Banach space is a Hilbert space? Valerio Capraro 2011-10-07T09:50:28Z 2011-12-07T11:03:20Z <p>In this simple note <a href="http://arxiv.org/abs/0907.1813" rel="nofollow">http://arxiv.org/abs/0907.1813</a> (to appear in Colloq. Math.), Rossi and I proved a characterization in terms of "inversion of Riesz representation theorem".</p> <p>Here is the result: let $X$ be a normed space and recall Birkhoff-James ortogonality: $x\in X$ is orthogonal to $y\in X$ iff for all scalars $\lambda$, one has $||x||\leq||x+\lambda y||$.</p> <p>Let $H$ be a Hilbert space and $x\rightarrow f_x$ be the Riesz representation. Observe that $x\in Ker(f_x)^\perp$, which can be required using Birkhoff-James orthogonality:</p> <p><strong>Theorem:</strong> Let $X$ be a normed (resp. Banach) space and $x\rightarrow f_x$ be an isometric mapping from $X$ to $X^*$ such that</p> <p>1) $f_x(y)=\overline{f_y(x)}$</p> <p>2) $x\in Ker(f_x)^\perp$ (in the sense of Birkhoff and James)</p> <p>Then $X$ is a pre-Hilbert (resp. Hilbert) space and the mapping $x\rightarrow f_x$ is the Riesz representation.</p>