An example of norms that are not equivalent

In summary, there are two norms that can be used on a vector space that are not equivalent: the sup norm and L1 norm on the space of continuous functions on the unit interval, C[0,1]. These norms are not equivalent even though norms on a finite dimensional vector space are typically equivalent. The vector space must be infinitely dimensional for these norms to be not equivalent.
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Homework Statement


Give an example of two norms on a vector space that are not equivalent.


Homework Equations





The Attempt at a Solution



Hi everyone,

I know the definition for equivalent norms. I also know that norms on a finite dimensional vector space are equivalent. So clearly the vector space has to be infinitely dimensional.
In which case can you take the normal euclidean norm and the max norm? And if not, could someone please point me in the right direction?

Thanks for any help
 
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  • #2
Consider C[0,1] - continuous functions on the unit interval. Try the sup norm and L1 norm.
Of course there are a lot of different examples, depending on what stuff you are comfortable with.
 

What is an example of norms that are not equivalent?

An example of norms that are not equivalent is the cultural norm of individualism versus collectivism. In individualistic cultures, emphasis is placed on individual freedom, self-expression, and personal achievement. In contrast, collectivistic cultures prioritize group harmony, interdependence, and conformity to societal expectations.

How do norms that are not equivalent impact society?

Norms that are not equivalent can lead to misunderstandings, conflicts, and even discrimination between individuals or groups from different cultures. They can also shape the values and beliefs of a society and influence its social, economic, and political systems.

Can norms that are not equivalent change over time?

Yes, norms that are not equivalent can change over time as societies evolve and adapt to new circumstances. For example, as globalization and technological advancements continue to blur cultural boundaries, some societies may shift towards more individualistic norms while others may become more collectivistic.

What factors contribute to the development of norms that are not equivalent?

Several factors can contribute to the development of norms that are not equivalent, including historical events, geographic location, religion, and social values. These factors can shape a society's worldview and influence its perception of what is considered "normal" or acceptable behavior.

How can understanding norms that are not equivalent be beneficial?

Understanding norms that are not equivalent can help promote cultural sensitivity and respect for diversity. It allows individuals to better navigate and communicate with people from different backgrounds, and can lead to more inclusive and harmonious societies.

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