View source for Smooth space

A [[Normed space|normed space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s0859101.png" /> in which for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s0859102.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s0859103.png" /> there exists a unique functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s0859104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s0859105.png" />. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s0859106.png" /> is smooth if and only if its norm has a [[Gâteaux differential|Gâteaux differential]] at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s0859107.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s0859108.png" />.



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s0859109.png" /> be a solid (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591010.png" /> has a non-empty interior) convex set in a real linear topological space. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591011.png" /> is a support point if there is a hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591012.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591014.png" /> is totally contained in one of the two half-spaces determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591015.png" />. A support point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591016.png" /> is smooth (and called a smooth point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591017.png" />) if there is only one closed hyperplane supporting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591018.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591019.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591020.png" /> is smooth if every boundary point is smooth. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591021.png" /> is smooth, or smoothly normal, if the unit ball is smooth. Every separable Banach space can be smoothly renormed, i.e. there exists an equivalent smooth norm.

The dual property to  "smooth"  is strictly convex: Any non-identically zero continuous functional takes a maximum value on the closed unit ball at most at one point, or, equivalently, distinct boundary points of the closed unit ball have distinct supporting hyperplanes. For a linear normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591022.png" /> one has that if the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591023.png" /> is smooth (respectively, strictly convex), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085910/s08591024.png" /> is strictly convex (respectively, smooth).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.B. Holmes,   "Geometric functional analysis and its applications" , Springer  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V. Barbu,   Th. Precupanu,   "Convexity and optimization in Banach spaces" , Reidel  (1986)</TD></TR></table>