Prove that the ∞norm on ℝ^2 does not come from an inner product.

Name: Problem 14, Chapter 3: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2021-04-22T03:53:44-08:00
Duration: 13 min 18 s
Channel: Chris Trentman
Description: Prove that the ∞norm on ℝ^2 does not come from an inner product.

Prove that the ∞norm on ℝ^2 does not come from an inner...

Key Concepts

Norm

A norm is a function that assigns a nonnegative length or size to all vectors in a vector space and satisfies certain properties such as positivity, homogeneity, and the triangle inequality. Norms provide a way to measure distances and convergence in vector spaces and are foundational in functional analysis and related fields.

Inner Product

An inner product is a function that takes two vectors and returns a scalar, satisfying properties like linearity, symmetry (or conjugate symmetry in the complex case), and positive-definiteness. Inner products induce norms via the formula ||v|| = sqrt(<v, v>), and they allow for geometric concepts such as angles and orthogonality in vector spaces.

Parallelogram Law

The parallelogram law is a property that must be satisfied by any norm induced by an inner product. It states that for all vectors u and v, the sum of the squares of the norms of u+v and u?v equals twice the sum of the squares of the norms of u and v. This law is both necessary and sufficient for a norm to arise from an inner product, and checking it is a common method to determine if a given norm is inner product-induced.

Counterexample via the Infinity Norm

The infinity norm on ?², defined as the maximum absolute value of its components, can be shown not to satisfy the parallelogram law. By choosing specific vectors, one can demonstrate that the equality required by the parallelogram law fails. This violation implies that the infinity norm cannot be derived from any inner product, underscoring an important distinction between general norms and inner product-induced norms.

Transcript

00:01 In part a, we're asked to prove that the one norm is in fact a norm on rn.

00:11 To do this, we will prove this directly from the definition of norm.

00:16 It would be wise to learn what the axioms of a norm are.

00:28 We want to see if the one norm satisfies the norm axioms.

00:34 Let's check norm axiom number one.

00:39 So let v be a vector in rn.

00:44 In other words, v is equal to a1 through a1 through an.

01:00 Hunter, hunter.

01:04 Of course, by definition, the one norm of v is the maximum of the absolute value of the components ai.

01:22 Suppose that v is a non -zero vector.

01:30 Then very clearly, it follows with the norm of v1, which is the maximum of the ai in absolute value.

01:45 Well, since the norm of v is non -zero, that means that there exists an a -i such that a -i is non -zero.

01:53 And therefore the absolute value of ai is also non -zero.

02:02 In fact, it's strictly greater than zero.

02:06 And so this norm is going to be greater than or equal to, actually i'll call this a j, this is greater than or equal to the absolute value of a j, which is greater than zero.

02:23 Suppose that v equals zero instead.

02:26 So we have that the one norm of the zero, vector well this is the maximum of the absolute value of each of the components which is zero which is just zero therefore we've shown that property and one holds for the one norm let's check at the second axiom holds so let k be a real number scalar.

03:14 Well, in the norm of k times v, this is the one norm.

03:21 This is the one norm of k times a1, all the way up through k times a n, which is the maximum of the absolute value of the k times a i's, which this is the same as the maximum of the maximum of the absolute value of k times the absolute value of ai, which of course this is the same as the absolute value of k times the maximum of the absolute value of ai, which is the same as the absolute value of k times the norm of v.

04:05 Therefore it follows that property n2 also holds for the one norm on our end.

04:18 Finally, we'll prove property n3.

04:20 This is the hardest property, the triangle inequality.

04:26 So we'll let, well, u, this will have components a1, a2 up through an, and v have components b1b2 all the way up to bn.

04:44 Now consider the one norm of u plus v.

04:50 This is equal to the one norm of a1 plus b1 all the way up to a .n plus bn, which is equal to the maximum of the absolute values of the ais plus bis.

05:13 And of course, by triangle inequality for real numbers, this is less than equal to the maximum of the absolute value of a .i.

05:23 Plus the absolute value of b .i.

05:28 And then a property of maximums is that the maximum of the sum of two positive numbers is going to be less than or equal to the maximum of the first number plus the maximum over the second set of positive numbers.

05:53 So this is less than or equal to the maximum of the absolute value of ai plus the maximum of the maximum of the absolute value of a .i.

06:03 Plus the maximum of the absolute value vi over all the i's, which this is clearly the one norm of u plus the one norm of v, and therefore property n3 holds for the one norm as well.

06:29 Since satisfies properties n1n to an n3, it follows that the one norm is in fact a norm, on the vector space rn.

06:46 Okay.

06:50 Then in part b, where i likewise has to show that the infinity norm is a norm on rn.

06:58 Once again, to do this, we'll do it directly using the axioms of the norm.

07:06 So, i made a mistake, actually.

07:15 When i smoke it, i feel...

07:18 In part a, this was actually supposed to be the infinity norm.

07:22 So let's make this part b instead.

07:27 This should be infinity instead of one...