Banach–Mazur rotations problem

In the ordinary plane, rotating moves the unit vector (1, 0) to (3/5, 4/5) while preserving every length. Both have length one. A whole-space symmetry does more than move those two points: it must preserve lengths and vector addition everywhere.

Can an infinite-dimensional real space allow such an onto linear symmetry from any unit vector to any other, yet use a length rule that does not come from an inner product, a generalized dot product? The target restricts to separable Banach spaces, where limits exist for Cauchy sequences and a countable set can approximate every point.[1]

Can whole-space linear motions take any unit vector to any other while preserving every length, without those lengths coming from a dot product? In the ordinary plane, a rotation sends $(1,0)$ to $(3/5,4/5)$; the latter has squared length $9/25+16/25=1$. This finite-dimensional illustration teaches the symmetry, not an unresolved example.

A norm assigns nonnegative lengths, is zero only at zero, scales by the absolute scalar, and satisfies the triangle inequality. A Banach space is complete for its resulting distance: Cauchy sequences converge in the space. Separable means a countable subset approximates every point arbitrarily closely. The target uses real scalars and infinite dimension.

For every two unit vectors $u,v$, assume there is a surjective linear isometry $T:X\to X$ with $T(u)=v$. Such a map preserves vector addition, scalar multiplication and the norm everywhere. Must the given norm satisfy $\|x\|^2=\langle x,x\rangle$ for an inner product? A different norm making the space Hilbert would not establish this claim.[1]

For spaces with finitely many independent coordinate directions, moving any unit vector to any other by an onto linear length-preserving map is known to force an inner-product length rule. The cited paper also proves a result with a stronger symmetry requirement involving triples of points. Neither covers the target: a separable infinite-dimensional space with only the stated ability to move single unit vectors.[1]

The paper’s stronger three-point theorem does not decide this target.[1]

An abstract length rule need not begin with angles or a dot product. Whole-space symmetries provide another way to probe its geometry. Asking what that symmetry forces connects the group of allowed motions with how lengths of sums of vectors fit together.[1]

The connection can be tested through lengths of sums and differences. An inner-product norm obeys the parallelogram identity $\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2$. A norm alone need not obey it. For the plane’s maximum-coordinate norm and $x=(1,0),y=(0,1)$, the left side is two while the right side is four.

The square unit ball of that norm also has different kinds of unit points: a corner cannot be sent to an edge midpoint by an onto linear symmetry of the ball, because such a map preserves extreme points. The example illustrates how geometry constrains symmetry; it is finite-dimensional and fails the required transitivity. The open problem asks whether the corresponding symmetry information in the specified infinite-dimensional class still forces inner-product geometry.[1]