Am I understanding "supremum" correctly

[pyroknife]

If let's say I have an expression:
$x\leq y$

Since the supremum is defined as the "least upper bound," does this make sup(x) for this case $x=y$ or is it $x = \infty$?

 

[Hornbein]

If let's say I have an expression:
$x\leq y$

Since the supremum is defined as the "least upper bound," does this make sup(x) for this case $x=y$ or is it $x = \infty$?

The least upper bound of what? x or y? If both are free variables then it is rather useless. A "least upper bound" is the smallest number that is greater than or equal to every number in a set. In most of the interesting cases the set is infinite.

 

[pyroknife]

The least upper bound of what? x or y? If both are free variables then it is rather useless. A "least upper bound" is the smallest number that is greater than or equal to every number in a set. In most of the interesting cases the set is infinite.

In this case I am asking for the supremum of x, sup(x), which would be the least upper bound of x. Would it be x = y?

 

[PAllen]

The least upper bound of what? x or y? If both are free variables then it is rather useless. A "least upper bound" is the smallest number that is greater than or equal to every number in a set. In most of the interesting cases the set is infinite.

Maybe fixed after you commented, but OP reads sup(x). Then, sup(x)=y is correct for the set of x ≤ y, assuming y is a given value. Of course, this is a totally trivial case.

 

[PAllen]

In this case I am asking for the supremum of x, sup(x), which would be the least upper bound of x. Would it be x = y?

You wouldn't say x=y, you would say sup(x)=y, for the specified set.

 

[pyroknife]

Maybe fixed after you commented, but OP reads sup(x). Then, sup(x)=y is correct for the set of x ≤ y, assuming y is a given value. Of course, this is a totally trivial case.

Thanks. If instead we want sup(y), would that be infinity?

I am asking this question as a follow up of another thread I made right below this one regarding norms and supremums.

 

[pyroknife]

You wouldn't say x=y, you would say sup(x)=y, for the specified set.

ah yes, thanks

 

[WWGD]

Thanks. If instead we want sup(y), would that be infinity?

I am asking this question as a follow up of another thread I made right below this one regarding norms and supremums.

But as you wrote it, it seems y is just one number, and not a set, so, trivially, sup{y}=y (on most "reasonable" choices of orderings , where $a \leq a$)

 

[PAllen]

Thanks. If instead we want sup(y), would that be infinity?

I am asking this question as a follow up of another thread I made right below this one regarding norms and supremums.

That depends on how you define the set against which x≤y is defined. That is, given x, and the set of y such that x ≤ y of real numbers, then sup(y) does not exist. If, instead, you use the set of extended reals that includes +/- ∞ as wellas the conventional reals, then, indeed, sup(y)=∞.

 

[pyroknife]

That depends on how you define the set against which x≤y is defined. That is, given x, and the set of y such that x ≤ y of real numbers, then sup(y) does not exist. If, instead, you use the set of extended reals that includes +/- ∞ as wellas the conventional reals, then, indeed, sup(y)=∞.

But as you wrote it, it seems y is just one number, and not a set, so, trivially, sup{y}=y (on most "reasonable" choices of orderings , where $a \leq a$)

I see. This was not a very good hypothetical example. This is a question in direct relation with another thread of mine, https://www.physicsforums.com/threads/is-this-proof-valid-of-an-infty-norm-valid.854189/
where I am considering the infinite matrix norm, who's definition involves a supremum over a set of numbers.