Kunugui's theorem

The set of Kunugui says that every metric space isometric to a Banach space can be embedded.

formulation

Be a metric space. If is empty, it can be embedded trivially, otherwise a fixed point is chosen. For each one is now explained by a real function . Then the mapping is an isometry of into the Banach space of bounded functions . ${\displaystyle (X,d)}$ ${\displaystyle X=\emptyset }$ ${\displaystyle X}$ ${\displaystyle s\in X}$ ${\displaystyle x\in X}$ ${\displaystyle f_{x}(y):=d(x,y)-d(y,s)}$ ${\displaystyle X}$ ${\displaystyle x\mapsto f_{x}}$ ${\displaystyle (X,d)}$ ${\displaystyle B(X)=\{f\colon X\to \mathbb {R} \mid f{\mbox{ beschränkt}}\}}$

Remarks

The above statement consists of two parts, on the one hand it must be shown that they are all restricted (with regard to the supreme norm ) and that the assignment is actually an isometry. Both follow from the reverse triangle inequality . It applies by definition ${\displaystyle f_{x}}$ ${\displaystyle x\mapsto f_{x}}$

${\displaystyle \|f_{x}\|=\sup _{y\in X}|f_{x}(y)|=\sup _{y\in X}|d(x,y)-d(y,s)|}$.

According to the triangle inequality, the last expression is at most, and since it is fixed, it is bounded. In addition, for two points , ${\displaystyle d(x,s)}$ ${\displaystyle s\in X}$ ${\displaystyle f_{x}}$ ${\displaystyle x,x'\in X}$

${\displaystyle \|f_{x}-f_{x'}\|=\sup _{y\in X}|(d(x,y)-d(y,s))-(d(x',y)-d(y,s))|=\sup _{y\in X}|d(x,y)-d(x',y)|}$.

The last term is highest and if you look for z. B. inserts the point , you can see that even the equality holds. ${\displaystyle d(x,x')}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle \|f_{x}-f_{x'}\|=d(x,x')}$

The remarkable thing about Kunugui's theorem is the simple idea of subtracting the term from the intuitively obvious distance , and thus to achieve the limitation of the mapping . ${\displaystyle d(x,y)}$ ${\displaystyle d(y,s)}$ ${\displaystyle f_{x}\colon X\to \mathbb {R} }$

The fact that a metric space can be isometrically embedded in a complete room does not mean that it is complete itself. For example, the space with the Euclidean metric is incomplete - among other things, the Cauchy sequence does not converge - but it can still be isometrically embedded in the complete space through the inclusion . ${\displaystyle \mathbb {R} \setminus \{0\}}$ ${\displaystyle (1/n)_{n\geq 1}}$ ${\displaystyle \mathbb {R} }$

literature

• Kinjirô Kunugui: Applications des spaces à une infinité de dimensions à la théorie des ensembles. In: Proceedings of the Imperial Academy. 11, 9, 1935, ISSN  0369-9846 , pp. 351-353.