Properties of pointwise scalar multiplication of sets in normed spaces. #
We explore the relationships between scalar multiplication of sets in vector spaces, and the norm. Notably, we express arbitrary balls as rescaling of other balls, and we show that the multiplication of bounded sets remain bounded.
If s
is a bounded set, then for small enough r
, the set {x} + r • s
is contained in any
fixed neighborhood of x
.
In a real normed space, the image of the unit ball under scalar multiplication by a positive
constant r
is the ball of radius r
.
In a real normed space, the image of the unit closed ball under multiplication by a nonnegative
number r
is the closed ball of radius r
with center at the origin.
In a nontrivial real normed space, a sphere is nonempty if and only if its radius is nonnegative.
Any ball metric.ball x r
, 0 < r
is the image of the unit ball under λ y, x + r • y
.
Any closed ball metric.closed_ball x r
, 0 ≤ r
is the image of the unit closed ball under
λ y, x + r • y
.