Operator on polynomials, antiderivativeWe are given a linear map: R [ X ] ∋ p...

Emmy Knox

Emmy Knox

Answered

2022-06-26

Operator on polynomials, antiderivative
We are given a linear map:
R [ X ] p q R [ X ] ,     q = p ,     q ( 0 ) = 0 and two norms on R [ X ] : | | p | | = sup t [ 0 , 1 ] | p ( t ) | , | | p | | 1 = 0 1 | p ( t ) | d t.
I want to check whether the map is continuous in these norms.
When it comes to the first norm, I get:
| | q | | = sup t [ 0 , 1 ] | q ( t ) |
By Lagrange mean value theorem we have that there exists a [ 0 , 1 ] such that q ( a ) = q ( t ) q ( 0 ) t , so q ( t ) sup [ 0 , 1 ] | q ( t ) | .
And | | p | | = sup t [ 0 , 1 ] | p ( t ) | = sup t [ 0 , 1 ] | q ( t ) | | | q | |
Is that correct so far?

Answer & Explanation

Punktatsp

Punktatsp

Expert

2022-06-27Added 22 answers

Step 1
Your work is correct, but in my opinion this makes things more clear:
q ( x ) = 0 x p ( t ) d t .
Then, for the norm
| q ( x ) | 0 x | p ( t ) | d t x p p .
Step 2
Fot the 1 norm 0 1 | q ( x ) | d x = 0 1 | 0 x p ( t ) d t | d x 0 1 0 1 | p ( t ) | d t d x = p 1 .

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