Let ·1 and ·2 be two different norms on a vector space V. (a) Prove that 𝐯=max{𝐯1,𝐯2} defines a

Let ·1 and ·2 be two different norms on a vector space V....

Key Concepts

Geometric Mean of Norms

The geometric mean, defined as the square root of the product of two norms, does satisfy nonnegativity and is zero only for the zero vector. However, it may fail to satisfy the triangle inequality. The product structure inherent in the geometric mean does not lend itself to straightforward manipulation required by the triangle inequality, making it an unreliable candidate for a norm under general circumstances.

Arithmetic Mean of Norms

The arithmetic mean of two norms is defined by averaging their values. This construction preserves the norm axioms because linear combinations of norms (with nonnegative coefficients that sum to one) maintain both homogeneity and the triangle inequality. Since each individual norm is positive and satisfies the triangle inequality, their weighted average, such as the arithmetic mean, will naturally inherit these properties, ensuring it forms a valid norm.

Maximum of Two Norms

Taking the maximum of two norms defined on the same vector space yields a function that also satisfies the properties of a norm. The nonnegativity and absolute scalability follow directly from the corresponding properties of each individual norm, while the triangle inequality is verified by applying the maximum function and the triangle inequality from each norm. This construction is useful because it often retains the necessary control over the size of vectors while merging the characteristics of different norm measures.

Norm

A norm on a vector space is a function that assigns a nonnegative real number to each vector in a way that captures the intuitive notion of length. It must satisfy three main properties: positive definiteness (the norm is zero only for the zero vector), absolute scalability (scaling the vector scales the norm by the absolute value of the scalar), and the triangle inequality (the norm of the sum of two vectors is at most the sum of the norms). These properties are critical for the structure they provide in analysis and geometry.

Minimum of Two Norms

Defining a function as the minimum of two norms can fail to satisfy one or more of the norm axioms, particularly the triangle inequality. Although the minimum of nonnegative values is nonnegative and respects zero only at the zero vector, it might not preserve the triangle inequality. This is because for some vectors the minimum may give artificially small values that do not accommodate the sum of vectors, making this construction generally unsuitable as a norm.

Transcript

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

00:14 We do this, we're going to prove this directly using the norm axioms.

00:20 So, let k be a real number and we'll let f and g be continuous function.

00:38 On the interval ab.

00:41 To prove property n1, well, suppose that f is a non -zero function, this means that there exists some s in ab such that f of s is non -zero.

01:10 We'll recall what the one norm of a function is a function is is it's the integral from a to b of the absolute value of f of t d t.

01:38 And we know from before that the absolute value of f of s is since non -zero is greater than zero.

01:47 We know from a theorem in calculus that if you have a continuous function and it is positive at one point, then its integral over that interval is in fact positive.

02:09 This is since the absolute value of f is continuous on ab and absolute value of f of f is positive.

02:24 Likewise, we have the one norm of the zero function, well, i guess be more accurate.

02:35 The one norm of the zero function, this is the integral from a to b of the absolute value of, well, 0 of t, dt, which is the same as the integral from a to b of just 0, dt, which is just zero.

02:57 Therefore, it follows that axiom n1 holds for the one norm.

03:08 On the set of continuous functions on ab.

03:15 Okay, let's check n2.

03:18 Well, we have the one norm of k times f.

03:22 Well, this is the interval from a to b of the absolute value of, well, k times f of t, dt, which is the same as the integral from a to b of by definition, scale of times a function, this is k times f of t d t, and then this is equal to the interval from a to b of the absolute value of k times the absolute value of f of t d t by homogeneity of integrals.

04:01 This is the absolute value of k times the integral from a to b of the absolute value of f of t d t which is clearly the absolute value of k times the one norm.

04:12 Of f.

04:15 Therefore, it follows that property n2 holds as well for the one norm on the set of continuous functions on the interval ab.

04:31 At last, we want to show that the third axiom, the triangle inequality holds.

04:37 This is the hardest one.

04:39 Well, we have the one norm of f plus g.

04:45 First of all, because f and g are continuous functions on ab, it follows that f plus g is a continuous function on ab.

04:56 I guess we should also point this out in our previous part that because k is real, follows that k times f is a continuous function on ab.

05:12 Just a little technical point to add, and therefore the one norm of f plus g is defined, and is given by the integral from a to b of the absolute value of f plus g of t dt.

05:37 By the definition of function addition, this is the integral from a to b of the absolute value of f of t plus g of t, dt.

05:49 Then by the triangle inequality for real number of.

05:59 This is less than equal to the integral from a to b of the absolute value of f of t plus the absolute value of g of t d t.

06:11 And then by addativity we can write this as the integral from a to b of the absolute value of f of t d t plus the integral from a to b of the absolute value of g of t, dt.

06:23 And clearly this is the one norm of f plus the one norm of g.

06:31 Therefore, it follows that axiom n3 holds for the one norm on the set of continuous functions on ab as well...

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