closed ball / metric spaces / bounded functions

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Sam Horton - 19 Oct 2008 16:57 GMT

Hello All,

When reading through my lecture notes I've come across something I don't
understand and would greatly appreciate if someone could describe the
following object to me.

# The closed ball of radius 1 in B[0,1] with center the function f(x)=x^2;
where B[0,1]:="the space of bounded functions from [0,1] to Reals" #

From the little I know I would think this is the set {g in B[0,1]: p(g(x),
f(x)=x^2) =< 1}where I guess p is some unspecified metric but this still
seems very unclear.

Cheers.

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Brian M. Scott - 19 Oct 2008 17:14 GMT

On Sun, 19 Oct 2008 16:57:51 +0100, Sam Horton
<no-email@anti-spam.org> wrote in
<news:2OIKk.40093$N11.25215@newsfe13.ams2> in
alt.math.undergrad:

> Hello All,

> When reading through my lecture notes I've come across something I don't
> understand and would greatly appreciate if someone could describe the
> following object to me.

> # The closed ball of radius 1 in B[0,1] with center the function f(x)=x^2;
> where B[0,1]:="the space of bounded functions from [0,1] to Reals" #

> From the little I know I would think this is the set {g in B[0,1]: p(g(x),
> f(x)=x^2) =< 1}where I guess p is some unspecified metric but this still
> seems very unclear.

Most likely B[0, 1] is being equipped with the sup norm: for
any f in B[0, 1], ||f|| = sup{|f(x)| : x in [0, 1]}. If so,
the metric p is simply the norm of the difference, and the
ball in question is {g in B[0, 1] : ||g - f|| <= 1}, where
f(x) = x^2. This expands to

{g in B[0, 1] : sup{|g(x) - x^2| : x in [0, 1]} <= 1}.

Pictorially it consists of all functions in B[0, 1] whose
graphs lie within the curved strip bounded above by
y = x^2 + 1 and below by y = x^2 - 1, and on the left and
right by x = 0 and x = 1.

Brian

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Math Forum / Mathematics / Undergraduate Math / October 2008

closed ball / metric spaces / bounded functions