Reports of the Academy of Sciences of the USSR

1. Volume 141, No. 5

MATHEMATICS

S. S. RYSHKOV

ON A CERTAIN MAPPING OF HILBERT SPACE INTO ITSELF

(Presented by Academician P. S. Aleksandrov, 21 VII 1961)

In the present note we construct a uniformly continuous in both directions homeomorphism of Hilbert space \(H\) onto a certain subset of this space, all points of which have nonnegative coordinates less than or equal to one. (Hilbert space is assumed to be given in some orthonormal basis.)

1. Let an orthonormal basis \(\{e_1, e_2, \ldots, e_r, \ldots\}\) be given in the Hilbert space \(H\). Consider the subspace \(H_2\) of the space \(H\) having as basis vectors the vectors \(\{e_2, e_4, \ldots, e_{2n}, \ldots\}\), and define the linear mapping \(\varphi: H \to H_2\), given on the basis vectors by the formula \(\varphi(e_k)=e_{2k}\). We note that any vector of the space \(H_2=\varphi(H)\) can be represented in a unique way as a linear combination with nonnegative coefficients of the vectors \(\{e_2, e_4, \ldots, e_{2n}, \ldots\}\) and \(\{-e_2, -e_4, \ldots, -e_{2n}, \ldots\}\) (under the condition that the vectors \(e_{2n}\) and \(-e_{2n}\) do not both occur with nonzero coefficients). If in this linear combination, instead of the vectors \(-e_{2k}\), we substitute the vectors \(e_{2k-1}\), then we obtain a mapping \(\psi: H_2 \to H\), whereby all coordinates of the points (vectors) belonging to the image of the space \(H_2\) are nonnegative. Combining the mappings \(\varphi\) and \(\psi\), we obtain a mapping \(\psi\varphi\) of the whole space \(H\) into a subset having nonnegative coordinates.

We note that the mutual continuity of the mappings \(\varphi\) and \(\psi\), as well as the uniformity of the homeomorphism \(\varphi\), are obvious, and we shall prove the uniformity of the homeomorphism \(\psi\).

Indeed, let \(M\) and \(N\) be two points of \(H_2\), and let the point \(M\) have coordinates
\((0, x_2, 0, x_4, 0, \ldots, x_{2k}, 0, \ldots)\), while \(N\) has coordinates
\((0, y_2, 0, y_4, 0, \ldots, y_{2k}, 0, \ldots)\); if for all \(k\) the signs of the numbers \(x_{2k}\) and \(y_{2k}\) coincide*, then the distance between the points \(M\) and \(N\) and the distance between the points \(\psi(M)\) and \(\psi(N)\) are the same. Now let \(\{x_{2k_1}, x_{2k_2}, \ldots, x_{2k_r}, \ldots\}\) be the collection of those coordinates of the point \(M\) whose sign coincides with the sign of the corresponding coordinates, by number, of the point \(N\). Consider the point \(P\) with coordinates
\((0, 0, \ldots, 0, x_{2k_1}, 0, \ldots, 0, x_{2k_2}, \ldots)\), and join the point \(P\) with the points \(M\) and \(N\). It is clear that the segment \(MP\) is perpendicular to the plane spanned by the vectors \(e_{2k_1}, e_{2k_2}, e_{2k_3}, \ldots, e_{2k_r}, \ldots\). Moreover, the triangle \(MPN\) has an obtuse angle at the point \(P\), since the scalar product of the vectors \(\overrightarrow{PM}\) and \(\overrightarrow{PN}\) is negative. Considering the scalar product of the vectors \(\overrightarrow{\psi(M)\psi(P)}\) and \(\overrightarrow{\psi(N)\psi(P)}\), we see that in the triangle \(\psi(M)\psi(P)\psi(N)\) the angle at the point \(\psi(P)\) is straight. From what has been said above it is clear that the segment \(MP\) is equal to the segment \(\psi(M)\psi(P)\), and the segment \(NP\) is equal to \(\psi(N)\psi(P)\). Denoting the distances \(MP, NP, MN\), and \(\psi(M)\psi(N)\), respectively, by \(\alpha, \beta, a\), and \(r\), and taking into account the remark about the angles of the triangles \(MNP\) and \(\psi(M)\psi(N)\psi(P)\), we have the relations

\[ \max(\alpha,\beta) \le a \le \alpha+\beta,\qquad r=\sqrt{\alpha^2+\beta^2}. \]

* We regard zero as having both the plus sign and the minus sign.

Hence

\[ \frac{\max(\alpha,\beta)}{\sqrt{\alpha^2+\beta^2}}\leq \frac{a}{r}\leq \frac{\alpha+\beta}{\sqrt{\alpha^2+\beta^2}} \]

or

\[ \frac{1}{\sqrt{2}}\leq \frac{a}{r}\leq \sqrt{2}. \]

From what has been said, the desired uniformity follows.

1. Bending of the space. Let us perform the mapping \(\varphi: H\to H\), constructed in item 1, and, for simplicity, renumber the basis vectors of the space \(H_2\) again consecutively, denoting them by \(\mathbf g_k\), and also renumber consecutively the basis vectors of the factor space \(H_1=H/H_2\) (the complement \(H_1\) of the space \(H_2\) in \(H\)), denoting them by \(\mathbf f_k\). (The coordinates in \(H_2\) are denoted by \(x_k\), in \(H_1\) by \(y_k\).) If to the basis of the space \(H_2\) we adjoin the vectors \(\mathbf f_{k_1}, \mathbf f_{k_2},\ldots,\mathbf f_{k_r}\), then we shall denote the corresponding space by \((H_2,\mathbf f_{k_1},\mathbf f_{k_2},\ldots,\mathbf f_{k_r})\).

We now define the mapping

\[ C_{i,n}^{k}=C_{i,n}^{k}:H_2\to (H_2,\mathbf f_k) \]

as follows. All coordinates of the point \(C_{i,n}^{k}(M)\), where \(M\in H_2\), except the \(i\)-th, coincide with the coordinates of the point \(M\). If the \(i\)-th coordinate of the point \(M\) is less than \(n\), then the \(i\)-th coordinate of the point \(C_{i,n}^{k}(M)\) also remains equal to the corresponding coordinate of the point \(M\), while the coordinate \(y_k\) along the vector \(\mathbf f_k\) is set equal to zero. If, however, the \(i\)-th coordinate \(x_i\) of the point \(M\) is greater than \(n\), then the \(i\)-th coordinate of the point \(C_{i,n}^{k}(M)\) is set equal to \(n\), and the coordinate \(y_k\) along the vector \(\mathbf f_k\) equal to \(x_i-n\). Figuratively, this mapping may be represented as a bending of the space \(H_2\) along the plane \(x_i=n\) at a right angle toward itself in the direction of the vector \(\mathbf f_k\); therefore we call it a bending \(C_{i,n}^{k}\).

Suppose the space \(H_2\) has already been bent in the directions \(\mathbf f_{k_1},\mathbf f_{k_2},\ldots,\mathbf f_{k_r}\) along \((i_1,n_1),(i_2,n_2),\ldots,(i_r,n_r)\), and no \(k_s\) and no pairs \((i_s,n_s)\) coincide with one another; then define the bending:

\[ C^* = C_{i,n}^{k}: C_{i_r,n_r}^{k_r} C_{i_{r-1},n_{r-1}}^{k_{r-1}}\cdots C_{i_1,n_1}^{k_1}(H_2) \to (H_2,\mathbf f_{k_1},\ldots,\mathbf f_{k_{r-1}},\mathbf f_{k_r},\mathbf f_k) \]

(where \(k\ne k_s\) and \((i,n)\ne(i_s,n_s)\) for all \(s\)) as follows. If \(i\ne i_s\) for all \(s\), then the bending \(C^*\) is defined as the mapping induced by the bending

\[ C_{i,n}^{k}:(H_2,\mathbf f_{k_1},\ldots,\mathbf f_{k_r})\to (H_2,\mathbf f_{k_1},\ldots,\mathbf f_{k_r},\mathbf f_k) \]

(the coordinates \(y_s\) for \(1\leq s\leq r\) are unchanged).

If \(i=i_s\) for some marked \(s\), but \(n<n_s\) for all marked \(s\), then the mapping \(C^*\) is defined as in the preceding case; if, however, there is such a marked \(s\) that \(n_s<n\) (we assume \(n_s\) maximal among those possessing this property), then we define our bending \(C^*\) as the induced bending of the space \((H_2,\mathbf f_{k_1},\ldots,\mathbf f_{k_r})\) in the direction of the vector \(\mathbf f_k\), but now along the plane \(y_{k_s}=n_s-n\), i.e. by the bending \(C_{k_s,n_s-n}^{k}\).

Theorem. The mappings \(C_{i,n}^{k}C_{j,m}^{l}:H_2\to (H_2,\mathbf f_l,\mathbf f_k)\) and \(C_{j,m}^{l}C_{i,n}^{k}:H_2\to (H_2,\mathbf f_k,\mathbf f_l)\) coincide.

Let us note, first, that the spaces \((H_2,\mathbf f_k,\mathbf f_l)\) and \((H_2,\mathbf f_l,\mathbf f_k)\) coincide.

Suppose first that \(i=j\) and, for definiteness, \(m>n\). Take a point \(M\) whose \(i\)-th coordinate \(x_i\) is less than or equal to \(n\); then

\[ C_{j,m}^{l}C_{i,n}^{k}(M)= \]

\(= C_{i,n}^{k} C_{j m}^{l}(M)=M\), obviously. Next take a point \(M\) for which \(m \geqslant x_i>n\); then \(C_{j,m}^{l}(M)=M\) and \(C_{j,m}^{l}(C_{i,n}^{k}(M))=C_{i,n}^{k}(M)\), or, substituting the first equality into the right-hand side of the second, we have \(C_{j,m}^{l}(C_{i,n}^{k}(M))=C_{i,n}^{k}(C_{j,m}^{l}(M))\), i.e. \(C_{j,m}^{l}C_{i,n}^{k}(M)=C_{i,n}^{k}C_{j,m}^{l}(M)\).

There remains only the case \(x_i>m\); in this case the coordinate \(y_k\) with respect to the vector \(\mathbf f_k\) of the point \(C_{i,n}^{k}(M)\) is equal to \(x_i-n\), at the point \(C_{j,m}^{l}C_{i,n}^{k}(M)\) the coordinate \(y_k\) is equal to \(m-n\), and the coordinate \(y_l=x_i-m\); the coordinates of the point \(C_{i,n}^{k}C_{j,m}^{l}(M)\) are examined in exactly the same way, which gives us the proof of the theorem if \(i=j\).

If, however, \(i\ne j\), one can consider the very same cases and verify the validity of our theorem. For example, if \(x_i>n\) and \(x_j>m\), where \(x_i\) and \(x_j\) are coordinates of the point \(M\), then the \(i\)-th coordinates both of the point \(C_{i,n}^{k}C_{j,m}^{l}(M)\) and of the point \(C_{j,m}^{l}C_{i,n}^{k}(M)\) are equal to \(n\); the \(j\)-th coordinates of the same points are equal to \(m\), respectively, \(y_k=x_i-n\), \(y_l=x_j-m\), for both the one and the other point, and, since the remaining coordinates of the point \(M\) are unchanged under our mappings, it follows that the points \(C_{j,m}^{l}C_{i,n}^{k}(M)\) and \(C_{i,n}^{k}C_{j,m}^{l}(M)\) coincide.

Let us note that from our theorem there follows the commutativity of any finite number of bendings.

3. Construction of the mapping \(C\). Number consecutively all pairs \((i,n)\) for \(i>0\), \(n>0\); if the pair \((i,n)\) has received the number \(k=k_{i,n}\), then denote the mapping \(C_{i,n}^{k}\) by \(C^{k}\), and construct the mapping
\(C=\ldots C^{k}\ldots C^{2}C^{1}: H_2\to (H_2,\mathbf f_1,\mathbf f_2,\ldots,\mathbf f_k,\ldots)=H\).
Let us record the obvious fact that the mapping \(C\) is a homeomorphism. Let us also note that the mapping \(C\) carries the whole set of points of the space \(H_2\) having nonnegative coordinates into a subset of the set of those points of the space \(H\) whose nonnegative coordinates are less than or equal to one. The latter is easy to see if one takes into account that each point \(M\in H_2\) has only a finite number of coordinates greater than one; consequently, only a finite number of the mappings \(C_{i,n}^{k}\) do not leave it fixed, i.e., in order to obtain the point \(C(M)\), it is enough to consider only a finite number of mappings \(C^{k}\), which we may consider in any order. If, for example, \(n+1>x_i\geqslant n\) and \(x_i\) is the first coordinate of the point \(M\) greater than one, then we first consider the mapping
\(C'=C_{i,n}^{k_{i,n}}C_{i,n-1}^{k_{i,n-1}}\ldots C_{i,1}^{k_{i,1}}: H_2\to (H_2;\mathbf f_{k_{i,1}},\mathbf f_{k_{i,2}},\ldots,\mathbf f_{k_{i,n}})\);
it is clear that the number of coordinates of the point \(C'(M)\) greater than one is one less than for the point \(M\). Similarly we arrive at the fact that all coordinates of the point \(C(M)\) are less than or equal to one.

The uniformity in both directions of the homeomorphism \(C: H_2\to H\) is established as in the case of the mapping \(\psi\).

4. It is obvious that the mapping we need is given by the formula \(C\varphi\psi\varphi: H\to H\).

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
11 V 1961