Does Scalar Multiplication Preserve the Supremum in Simple Sets?

Thread starter Seacow1988
Start date Feb 17, 2011
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Supremum

In summary, a Simple Supremum Question is a mathematical concept used to find the smallest upper bound of a set of numbers or functions. It is commonly used in scientific research and can be applied to real-world scenarios in fields such as economics, physics, and engineering. The difference between Simple Supremum and Maximum is that the Simple Supremum is the smallest upper bound while the Maximum is the largest value. To find the Simple Supremum, one must first determine the upper bound and then verify that it is greater than or equal to all other elements in the set.

Feb 17, 2011

Seacow1988

If S=sup {Sn: n>N}, is it true that kS= sup{kSn: n>N} where k is any scalar?

Thanks!

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Feb 17, 2011

Seacow1988

Woops. I meant k is any nonnegative scalar.

Thanks!

Feb 17, 2011

micromass

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Yes, I think that this is indeed true for nonnegative scalars k. The proof is not so hard: just show that S is an upper bound and that no lower upper bound exists.

What is a Simple Supremum Question?

A Simple Supremum Question is a mathematical concept that asks for the smallest upper bound of a set of numbers or functions. It is also known as the least upper bound or supremum.

How is a Simple Supremum Question used in science?

Simple Supremum Questions are commonly used in scientific research, particularly in areas such as optimization and analysis. They help determine the maximum or minimum value of a variable and are useful in making predictions and solving problems.

What is the difference between Simple Supremum and Maximum?

While the Simple Supremum is the smallest upper bound, the Maximum is the largest value in a set of numbers or functions. The Simple Supremum may or may not be equal to the maximum, depending on the set of values being evaluated.

Can Simple Supremum Questions be applied to real-world scenarios?

Yes, Simple Supremum Questions can be applied to real-world scenarios. They can be used to analyze data and make informed decisions in fields such as economics, physics, and engineering.

How do you find the Simple Supremum of a set of numbers or functions?

To find the Simple Supremum of a set of numbers or functions, you must first determine the upper bound by finding the maximum value. Then, you must check if the upper bound is the least upper bound by verifying that it is greater than or equal to every other element in the set. If it is, then the Simple Supremum is equal to the upper bound. If not, the Simple Supremum is the next largest number that is greater than all elements in the set.