How can I prove Minkowski's inequality for integrals?

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In summary, In order to prove Minkowski's inequality for integrals, we need to show that the integral of (f+g)^2 is less than or equal to the sum of the integrals of f^2 and g^2. To do this, we can use the Cauchy-Bunyakovsky-Schwarz inequality and the properties of Riemann integrals. This involves expanding the integral on the left side and using the Cauchy-Schwarz inequality to show that it is less than or equal to the sum of the integrals of f^2 and g^2. This can be achieved by using the inner product and induced norm, and applying the Cauchy-Schwarz inequality
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A_I_
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I need to proove the Minkowski's inequality for integrals.
I am taking a course in analysis.

[ int(f+g)^2 ] ^(1/2) =< [int(f^2)]^(1/2) + [int(g^2)]^(1/2)

now we are given that both f and g are Riemann integrable on the interval.
So by the properties of Riemann integrals, so is f^2,g^2 and fg.

We are also given a hint to expand the integral on the left and then use the Cauchy-Bunyakovsky-Schwarz inequality (now this I've already prooved in a previous exercice using the discriminant).

I was trying to expand the left side but i don't know what to do with the squared root, moreover i was trying to expand regardless the squared root and then at the end take a squared root but it still hasn't worked..

I need help =)
Thanks,

Joe
 
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  • #2
form what you wrote i assume you have an inner product given by

<f,g> = int fg dx

and the induced norm

|f| = <f,f>^½

so the Cauchy-Schwarz inequality is

|<f,g>| <= |f||g|

from what you get

<f,g> <= |f||g|

so starting with

|f+g| = <f+g.f+g> = (int (f+g)^2)^½ = (int (f^2+g^2+2fg)^½
= (int f^2 + int g^2 + int 2fg)^½ = (<f,f>+<g,g>+2<f,g>)^½
= (|f|^2+|g|^2+2<f,g>)^½

by cauchy-schwarz

<= (|f|^2+|g|^2+2|f||g|)^½ = [(|f|+|g|)^2]^½ = |f|+|g|

qed.
 
  • #3
i forgot a ^½ int the line

|f+g| = <f+g.f+g>

it should be

|f+g| = <f+g.f+g>^½
 

1. What is Minkowski's inequality?

Minkowski's inequality is a mathematical theorem that states that the sum of the p-th power of two real numbers is less than or equal to the p-th power of the sum of the two numbers, with p being any number greater than or equal to 1. In simpler terms, it is a way to compare the magnitude of two numbers when raised to a certain power.

2. Who discovered Minkowski's inequality?

Minkowski's inequality was discovered by German mathematician Hermann Minkowski in the late 1800s. He is also known for his contributions to the fields of geometry, number theory, and mathematical physics.

3. What is the significance of Minkowski's inequality?

Minkowski's inequality is significant because it has many applications in mathematics and physics. It is commonly used in functional analysis, probability theory, and the study of metric spaces. In physics, it is used to prove the Heisenberg uncertainty principle and to define the Minkowski space-time in special relativity.

4. What are the different forms of Minkowski's inequality?

Minkowski's inequality has three main forms: the integral form, the discrete form, and the vector form. The integral form involves the integral of two functions raised to the p-th power, the discrete form involves the sum of the p-th power of two sequences, and the vector form involves the p-norm of two vectors.

5. How is Minkowski's inequality related to other mathematical concepts?

Minkowski's inequality is closely related to other mathematical concepts such as the Cauchy-Schwarz inequality, Hölder's inequality, and the triangle inequality. It can also be extended to n numbers using the generalized Minkowski's inequality. Additionally, it has connections to other areas of mathematics such as convex analysis and functional analysis.

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