Smooth space

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A normed space $X$ in which for any point $x$ with $\|x\|=1$ there exists a unique functional $f\in X^*$ such that $f(x)=\|f\|=1$. A space $X$ is smooth if and only if its norm has a Gâteaux differential at all points $x$ with $\|x\|=1$.


Let $A$ be a solid (i.e. $A$ has a non-empty interior) convex set in a real linear topological space. A point $a\in A$ is a support point if there is a hyperplane $H$ passing through $a$ such that $A$ is totally contained in one of the two half-spaces determined by $H$. A support point $a\in A$ is smooth (and called a smooth point of $A$) if there is only one closed hyperplane supporting $a$ at $A$. The set $A$ is smooth if every boundary point is smooth. A space $X$ 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 $X$ one has that if the dual space $X^*$ is smooth (respectively, strictly convex), then $X$ is strictly convex (respectively, smooth).


[a1] R.B. Holmes, "Geometric functional analysis and its applications" , Springer (1975)
[a2] V. Barbu, Th. Precupanu, "Convexity and optimization in Banach spaces" , Reidel (1986)
How to Cite This Entry:
Smooth space. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.P. Vlasov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article