Smooth space

Smooth normalized spaces are examined in the mathematical sub-area of functional analysis. These are standardized spaces , the norm of which has a certain smoothness property .

Definitions

Let it be a standardized space, be the unit sphere and its edge , the so-called unit sphere . According to Hahn-Banach's theorem, there is a continuous , linear functional with and for each . ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle B_{X}:=\{x\in X;\,\|x\|\leq 1\}}$ ${\displaystyle S_{X}:=\{x\in X;\,\|x\|=1\}}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle f_{x}\in X^{*}}$ ${\displaystyle \|f_{x}\|=1}$ ${\displaystyle f_{x}(x)=1}$

This functional defines the hyperplane , the in cuts and no point on the inside contains the unit sphere. Such a hyperplane is called a hyperplane of support to , the functional is supporting functional to . If one imagines a hyperplane as a linear approximation of the spherical surface, it makes sense to call a point a smoothness point if there is exactly one supporting hyperplane , that is, if there is exactly one with and . ${\displaystyle f_{x}}$ ${\displaystyle \{y\in X;\,f_{x}(y)=1\}}$ ${\displaystyle B_{X}}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle f_{x}}$ ${\displaystyle x}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle x}$ ${\displaystyle f_{x}\in X^{*}}$ ${\displaystyle \|f_{x}\|=1}$ ${\displaystyle f_{x}(x)=1}$

A normalized space is called smooth if every point of the unit sphere is a smoothness point. The unit sphere of a smooth space is thus a smooth convex set .

Support picture

A figure is called a support figure if the following applies: ${\displaystyle f:X\setminus \{0\}\rightarrow X^{*}\setminus \{0\},x\mapsto f_{x}}$

• From follows ${\displaystyle \|x\|=1}$ ${\displaystyle \|f_{x}\|=f_{x}(x)=1}$
• For and applies . ${\displaystyle \lambda >0}$ ${\displaystyle x\in X\setminus \{0\}}$ ${\displaystyle f_{\lambda x}=\lambda f_{x}}$

By definition, there is exactly one support image in a smooth room, so one can speak of the support image of a smooth room. It can be shown that this norm-weak * -continuous, that is to say continuous, if one looks at the norm topology and the weak- * topology . ${\displaystyle X\setminus \{0\}}$ ${\displaystyle X^{*}\setminus \{0\}}$

Examples

The Euclidean norm on the left is smooth, the maximum norm on the right is not.

Two-dimensional space

Smoothness depends on the norm and can be lost when moving to an equivalent norm . This can already be seen in the example of two-dimensional space . If one provides the two-dimensional space with the Euclidean norm , the unit sphere is a circle and every point has exactly one supporting hyperplane, namely the tangent at this point, that is, it is smooth. Considering on the maximum norm , then the "unit sphere" a square . At every corner of the square there are infinitely many supporting hyperplanes, all other points are smoothness points. In order for a space to be smooth, every point of the unit sphere must be a point of smoothness, that is, it is not smooth. Since the Euclidean norm and the maximum norm on the are equivalent, you can see from this example that the smoothness can be lost when changing to an equivalent norm. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \|.\|_{2}}$ ${\displaystyle (\mathbb {R} ^{2},\|.\|_{2})}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \|.\|_{\infty }}$ ${\displaystyle (\mathbb {R} ^{2},\|.\|_{\infty })}$ ${\displaystyle \mathbb {R} ^{2}}$

Further examples

• Hilbert spaces are smooth, the support figure reads . ${\displaystyle x\mapsto f_{x}=\langle \cdot ,x\rangle }$
• The L ^p [0,1] -spaces and the sequence spaces are for smooth. More generally, evenly smooth spaces are smooth. ${\displaystyle \ell ^{p}}$ ${\displaystyle 1<p<\infty }$
• If a compact Hausdorff space with at least two points, then the function space of the continuous functions on with the supremum norm is not smooth. ${\displaystyle K}$ ${\displaystyle C(K)}$ ${\displaystyle K}$

Characterizations

The following statements about a normalized space are equivalent: ${\displaystyle (X,\|\cdot \|)}$

• ${\displaystyle (X,\|\cdot \|)}$ is smooth.
• The norm on is Gâteaux-differentiable , that is for each and exists . ${\displaystyle S_{X}}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle y\in X}$ ${\displaystyle \lim _{t\searrow 0}{\frac {\|x+ty\|-1}{t}}}$
• Every support map of the room is norm-weak * -continuous.
• There is a norm-weak * -continuous support map.
• For each and every sequence in with it follows that weakly * -converges. ${\displaystyle x\in S_{X}}$ ${\displaystyle (\varphi _{n})_{n\in \mathbb {N} }}$ ${\displaystyle S_{X^{*}}}$ ${\displaystyle \varphi _{n}(x)\rightarrow 1}$ ${\displaystyle (\varphi _{n})_{n\in \mathbb {N} }}$
• Every two-dimensional subspace is smooth.
• The orthogonality is right-additive, i.e. from and follows . ${\displaystyle x\perp y}$ ${\displaystyle x\perp z}$ ${\displaystyle x\perp (y+z)}$

duality

There is a close connection to strict convexity via duality .

• A normalized space is smooth if its dual space is strictly convex. ${\displaystyle (X,\|\cdot \|)}$
• A normalized space is strictly convex if its dual space is smooth. ${\displaystyle (X,\|\cdot \|)}$

The inversions generally do not hold.

Renormalizability

Since the smoothness can be lost in the transition to an equivalent norm, the question naturally arises as to which normalized spaces there are equivalent, smooth norms that become smooth through the transition to an equivalent norm. Such spaces are called smoothly renormalizable.

Reflexive spaces can be renormalized in a strictly convex manner and therefore, due to the above duality properties, can also be renormalized smoothly, even smoothly and at the same time strictly convexly renormalized. This applies more generally to rooms that are created to be weakly compact .

${\displaystyle \ell ^{\infty }}$ is not easy to renormalize.

Individual evidence

1. Jump up ↑ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Lecture Notes in Mathematics 485, Springer-Verlag (1975), ISBN 3-540-07402-3 , Chapter 2, §1
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Corollary 5.4.18
3. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Lecture Notes in Mathematics 485, Springer-Verlag (1975), ISBN 3-540-07402-3 , Chapter 2, §1, Theorem 1, formulated point by point for Banach spaces
4. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Theorem 5.4.19
5. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 5.4.21
6. ↑ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Lecture Notes in Mathematics 485, Springer-Verlag (1975), ISBN 3-540-07402-3 , Chapter 2, §1, Theorem 4, formulated for Banach spaces
7. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Lecture Notes in Mathematics 485, Springer-Verlag (1975), ISBN 3-540-07402-3 , Chapter 2, §1, Theorem 2, formulated for Banach spaces
8. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentences 5.4.5, 5.4.6
9. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Lecture Notes in Mathematics 485, Springer-Verlag (1975), ISBN 3-540-07402-3 , Chapter 5, §2, Corollary 2 on Theorem 2
10. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Lecture Notes in Mathematics 485, Springer-Verlag (1975), ISBN 3-540-07402-3 , Chapter 4, §5, Sentence 2