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Nonsurjective isometry need not preserve midpoints
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Mathematics
Area of mathematics
Analysis
Functional analysis
Normed vector space
Surjective isometry of normed vector spaces
Created
2026-09-24
Updated
2026-09-24
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Surjectivity in the
Mazur-Ulam theorem
is essential. The
map
u
:
R
⟶
(
R
2
,
∥
⋅
∥
∞
)
,
u
(
x
)
=
(
x
,
∣
x
∣
)
,
(1)
preserves all
distances
and fixes zero, but it is not linear and does not preserve every
midpoint
.
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Surjective isometry of normed vector spaces
Normed vector space
Functional analysis
Analysis
Area of mathematics
Mathematics
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(1)
Past exam of the mathematics course of the University of Cambridge
/
2023
/
ii
/
Paper 2
/
22F
/
b
/
Solution
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