Norms that are unit functions

[mahjk17]

How will I be able to do these problems?

For which norms is the constant function f(x) = 1 a unit function ?
a.) L^1 norm on R
b.) L^2 norm on R
c.) L^∞ norm on R
d.) L^1 norm on [-1,1]
e.) L^∞ norm on [-1,1]
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[daon2]

You'll have to define "unit function"

 

[mahjk17]

Sorry for the confusion it is a vector of length 1 in which these norms does f(x)= 1 have "norm" 1.

 

[daon2]

a and b should be obvious. Why?

For c, I'm honestly not sure.

In d, the L^1 norm of f

\(\displaystyle \displaystyle (\int_{-1}^1 1 dx)\). Does this equal 1?

In e, the L^oo norm of f will be 1.

\(\displaystyle \displaystyle \lim_{n\to\infty} \left(\int_{-1}^1 1^n dx \right)^{1/n} = \lim_{n\to\infty} 2^{1/n} = ?\)

The same will be true over any bounded interval.

 

[mahjk17]

I understand now, thank you very much !