Inner Product, Triangle and Cauchy Schwarz Inequalities

In summary: In this case, the norm is defined as the square root of the inner product, so it is necessary to use the square root in the calculation. Additionally, the norm satisfies the triangle inequality, so using ##<p + q, p + q>## would not prove anything.It is important to clarify the terminology and definitions before attempting the question. The triangle inequality states that the norm of the sum of two vectors is less than or equal to the sum of the norms of the individual vectors. The Cauchy-Schwarz inequality states that the inner product of two vectors is less than or equal to the product of their norms.To solve the problem, simply calculate the inner products
  • #1
Lelouch
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Homework Statement


1.png


Homework Equations


I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm (length) and distance between inner product. And some basic consequences of the inner product axioms for norm and distance, s.t. norm is greater equal to 0, norm multiplied by a constant, symmetry of distance, and distance is greater than or equal to 0.

The rest are examples of inner products i.e. on ##R^n##.

The Attempt at a Solution


I have solved the definite integral for the given ##p(x)## and ##q(x)## which has the value ##\frac{68}{21}##.

What I thought I need to show is that ##<p, q> \leq <p> + <q>## for the triangle inequality. But this seems nonsensical to me since the inner product is defined for two vectors and not for one.
Maybe I have to show ##<p, q> \leq <p, p> + <q, q>##? But at this point I am just guessing since I do not know the triangle inequality for inner products nor the Cauchy-Schwarz inequality for inner products.
 

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  • #2
Lelouch said:
Maybe I have to show ##<p, q> \leq <p, p> + <q, q>##? But at this point I am just guessing since I do not know the triangle inequality for inner products nor the Cauchy-Schwarz inequality for inner products.
Yes, an inner product defines an associated norm ||a|| 2= <a,a>. The triangle they are talking about is p, q, and p-q. To show that the norm satisfies the triangle inequality for that triangle, show that ##√<p-q, p-q> \leq √<p, p> + √<q, q>##.
 
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  • #3
FactChecker said:
Yes, an inner product defines an associated norm ||a|| 2= <a,a>. The triangle they are talking about is p, q, and p-q. To show that the norm satisfies the triangle inequality for that triangle, show that ##√<p-q, p-q> \leq √<p, p> + √<q, q>##.

I wanted to ask why the norm was defined as the inner product, but now right before I answered I noticed you added the ##\sqrt{}##.

It seems simple I just calculate the respective inner products using the integral, then square root and check whether the inequality holds for the numerical values.
However, why do we perform ##<p - q, p - q>## instead of ##<p + q, p + q>## as shown in Theorem 1(a) in the following link?http://mathonline.wikidot.com/the-triangle-inequality-for-inner-product-spaces
http://mathonline.wikidot.com/the-triangle-inequality-for-inner-product-spaces
 
  • #4
Lelouch said:
I wanted to ask why the norm was defined as the inner product, but now right before I answered I noticed you added the ##\sqrt{}##.

It seems simple I just calculate the respective inner products using the integral, then square root and check whether the inequality holds for the numerical values.
However, why do we perform ##<p - q, p - q>## instead of ##<p + q, p + q>## as shown in Theorem 1(a) in the following link?
http://mathonline.wikidot.com/the-triangle-inequality-for-inner-product-spaces
Since any of the three sides of a triangle is no longer than the sum of the lengths of the other two sides, both triangle examples work.
 
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  • #5
Lelouch said:

Homework Statement


View attachment 220858

Homework Equations


I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm (length) and distance between inner product. And some basic consequences of the inner product axioms for norm and distance, s.t. norm is greater equal to 0, norm multiplied by a constant, symmetry of distance, and distance is greater than or equal to 0.

The rest are examples of inner products i.e. on ##R^n##.

The Attempt at a Solution


I have solved the definite integral for the given ##p(x)## and ##q(x)## which has the value ##\frac{68}{21}##.

What I thought I need to show is that ##<p, q> \leq <p> + <q>## for the triangle inequality. But this seems nonsensical to me since the inner product is defined for two vectors and not for one.
Maybe I have to show ##<p, q> \leq <p, p> + <q, q>##? But at this point I am just guessing since I do not know the triangle inequality for inner products nor the Cauchy-Schwarz inequality for inner products.

Get the terminology straight first, before attempting the question. For any two vectors ##p,q##:
1. Triangle inequality: ##|| p + q || \leq || p || + || q ||##.
2. Cauchy-Schwartz: ##|\langle p,q \rangle | \leq || p || \cdot || q ||.##
Here, ##\langle \cdot , \cdot \rangle## is the inner product and ##||w|| = \sqrt{\langle w,w \rangle }## is the associated "norm".

In your case the easiest thing is to just do all the integrals and show that you get the above inequalities with the given ##p,q##.
 
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  • #6
Ray Vickson said:
Get the terminology straight first, before attempting the question. For any two vectors ##p,q##:
1. Triangle inequality: ##|| p + q || \leq || p || + || q ||##.
2. Cauchy-Schwartz: ##|\langle p,q \rangle | \leq || p || \cdot || q ||.##
Here, ##\langle \cdot , \cdot \rangle## is the inner product and ##||w|| = \sqrt{\langle w,w \rangle }## is the associated "norm".

In your case the easiest thing is to just do all the integrals and show that you get the above inequalities with the given ##p,q##.

Thank you. I was able to solve the problem given your explanation of the terminology.
 

Related to Inner Product, Triangle and Cauchy Schwarz Inequalities

1. What is an inner product?

An inner product is a mathematical operation that takes two vectors and returns a scalar value. It is denoted by <x, y> and is used to measure the angle between two vectors, as well as the length of a vector.

2. How is the inner product related to the triangle inequality?

The triangle inequality states that in any triangle, the sum of the lengths of any two sides is always greater than the length of the third side. In terms of inner products, this means that the inner product of any two vectors is always less than or equal to the product of their lengths.

3. What is the Cauchy-Schwarz inequality?

The Cauchy-Schwarz inequality is a mathematical rule that states that the inner product of two vectors is always less than or equal to the product of their lengths. In other words, it shows that the angle between two vectors cannot be larger than the product of their lengths.

4. How is the Cauchy-Schwarz inequality used in mathematics?

The Cauchy-Schwarz inequality is used in various branches of mathematics, such as linear algebra, functional analysis, and geometry. It is often used to prove other theorems and inequalities, and has applications in optimization, statistics, and physics.

5. Can the Cauchy-Schwarz inequality be extended to more than two vectors?

Yes, the Cauchy-Schwarz inequality can be extended to more than two vectors. In fact, it can be extended to any number of vectors, and the resulting inequality is known as the generalized Cauchy-Schwarz inequality. This inequality is used in higher-dimensional spaces and has various applications in mathematics and physics.

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