Consider the functions f(x)=1 and g(x)=x-(3)/(4) as elements of the vector space C^0[0,1]. For e

Consider the functions f(x)=1 and g(x)=x-(3)/(4) as elements...

Key Concepts

Triangle Inequality

The triangle inequality is a fundamental property of norms stating that the norm of the sum of two vectors is less than or equal to the sum of the norms of the individual vectors. This property is central to the concept of distance in normed spaces, ensuring that the 'direct route' between two points is always the shortest. In the case of function norms, it guarantees that the combined effect of two functions does not exceed the sum of their individual sizes.

Space of Continuous Functions (C^0[0,1])

The space C^0[0,1] consists of all continuous functions defined on the interval [0, 1]. This space forms a vector space, meaning that any linear combination of continuous functions is also continuous. When equipped with various norms such as L^p and L^?, this space becomes a normed vector space, allowing for the analysis and comparison of function sizes in a rigorous mathematical framework.

L^? Norm

The L^? norm, also known as the uniform or supremum norm, measures the maximum absolute value of a function over its domain. Instead of integrating over the domain like the L^p norms, it focuses on the worst-case scenario—the largest deviation of the function across the interval. This norm is particularly useful in contexts where uniform convergence or maximum deviation is of interest.

L^p Norms

The L^p norms are a family of norms defined for functions by taking the p-th power of the absolute value of the function, integrating it over the domain, and then taking the p-th root of the result. These norms generalize the concept of measuring length or size increasingly sensitively as p varies. For example, common cases include the L^1 norm (p = 1), L^2 norm (p = 2), and L^3 norm (p = 3), each providing a different measure of the function’s amplitude over an interval.

Normed Vector Space

A normed vector space is a vector space equipped with a function called a norm, which assigns a non?negative real number to each vector in the space. The norm quantitatively measures the 'size' or 'length' of vectors and satisfies properties such as absolute scalability, the triangle inequality, and being zero only for the zero vector. In the context of function spaces, these vectors are often functions defined on a domain, and their norms provide a way to measure their magnitude or deviation.

Transcript

00:01 In this video, we're going to go through the answer to question number 34 from chapter 10 .3.

00:08 So first, in part a, we're asked to show that the norm of f, of a function f is always great and equal to zero.

00:14 And that the norm of f is only ever zero.

00:16 If and only if, the function itself is equal to zero.

00:20 So to show this, we start with looking at the squared norm of f.

00:24 So by definition, that's equal to the integral between a and b of f squared times by the weight function.

00:31 Dx.

00:36 This weight function is always greater than zero.

00:39 A function squared is always going to be greater than zero.

00:43 So the integral, the whole of this integrand is going to be minimum when it's equal to zero.

00:53 Then the integral of zero is just going to be equal to zero.

01:01 So they're taking the square root of this, and this gives us that the norm of f is less than or equal, yeah, hang on a second.

01:14 Of course, this should just be greater than equal to.

01:17 So the norm of f is greater than or equal to zero.

01:23 So when is it precisely zero? well, the norm of f is precisely equal to zero if and only if the integrand is equal to zero, because that was the minimum point that we put in here.

01:39 And this intergrand is equal to well we know that we know that the function w is equal is greater than zero so therefore the function f squared must be equal to zero so therefore the function f must be equal to zero and that completes the proof of part a part b we have to show that the norm of a constant multiple of function is equal to a constant multiple of the norm of that function so to show that we start with the squared norm of a constant times by function.

02:23 So by definition, that's equal to the interval between a and b of a function cf squared of x times the weight function dx.

02:36 Rewriting this function, we have the c squared times by the function f squared.

02:43 This c square can come out to the front.

02:47 So we get c squared multiplied by the integral between a and b of f squared w d x which is just the squared norm of f take the square root of both sides and you get the answer okay part c we're asked to prove the triangle inequality for the norm so that is that the norm of f plus g is less than of equal to norm of f plus the norm of j so to do this we'll start by proving a proving a lemma which will be theorem on the way to proving the actual theorem that we want to prove.

03:32 So first let's take a look at this guy.

03:35 So what we've got here is the norm of a function and the function is one function minus some scalar multiple of another function.

03:49 So we know that the norm is just a real number, so this is just a a constant.

03:57 So using part a we know that this because it's a norm, is going to be greater than equal to 0.

04:07 Expanding this using just the definition of the norm, we've got that it's the integral between a and b of f minus c, g, where i've defined c to be this constant, all squared times by the weight function dx...

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