Normed Vector Space Homework: Prove No Inner-Product Norm

In summary, the vector space C[a,b] of all continuous complex-valued functions f(x), x \in [a,b] has a norm, which does not satisfy the parallelogram law.
  • #1
poobar
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Homework Statement



Consider the vector space C[a,b] of all continuous complex-valued functions f(x), x [tex]\in[/tex] [a,b]. Define a norm ||f|| sup = max{|f(x)|, x [tex]\in[/tex] [a,b]}. (Math Note: technically we want to use sup instead of max but a physicists operational definition of max is the mathematial notion of sup).

a. Show that this is a norm.
b. Show that this norm does not satisfy the parallelogram law, ||x-y|| + ||x+y|| = 2||x||[tex]^{2}[/tex] + 2||y||[tex]^{2}[/tex]. Therefore, it cannot be an inner-product norm

Homework Equations



IF ||v|| = 0 then |v> = 0
||v1 + v2|| [tex]\leq[/tex] ||v1|| + ||v2|| (triangle inequality)
||v|| [tex]\geq[/tex] 0
||av|| = |a| ||v|| if a [tex]\in[/tex] complex

The Attempt at a Solution



I really have no idea where to start on this. i tried to apply the rules of a norm, but i am very confused. please help.
 
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  • #2
Why do you have "no idea how to start". You wrote out what is required for a norm. What happens if you put your specific definition of the norm into those?

If max(|f(x)|)= 0, what can you say about f(x)?
For any given x, [itex]|f(x)+ g(x)|\le |f(x)|+ |g(x)|[/itex].
max (a g(x)), for fixed a, is equal to a(max(g(x)).

By the way, because [a, b] is a closed interval, even the mathematical definitions of "sup" and "max" are the same.
 
  • #3
ok, that makes sense. i guess the part i am confused about is how to go about proving that the definition fits all of the rules.

also, for the part about the parallelogram law, do i have to break f(x) into real and imaginary parts in oder to prove that it doesn't hold true for the rule? i feel like that's a dumb question, but this is part that is most confusing to me. since it is a complex valued function, this is the first thing that comes to mind.
 

1. What is a normed vector space?

A normed vector space is a mathematical concept that combines the concept of a vector space, which is a set of objects that can be added together and multiplied by scalars, with the concept of a norm, which is a function that assigns a non-negative value to each vector in the space.

2. What is an inner-product norm?

An inner-product norm is a type of norm that is defined using an inner product, which is a mathematical operation that takes two vectors and produces a scalar value. The inner-product norm is used to measure the length or magnitude of a vector in a normed vector space.

3. Why is it important to prove that there is no inner-product norm in a normed vector space?

Proving that there is no inner-product norm in a normed vector space is important because it helps to establish the limitations of the space and the types of norms that can be defined within it. It also helps to understand the properties and structure of the space in a more comprehensive manner.

4. How can one prove that there is no inner-product norm in a normed vector space?

To prove that there is no inner-product norm in a normed vector space, one must show that the properties of an inner product, such as symmetry, linearity, and positive-definiteness, cannot be satisfied by any function in the space. This can be done by providing a counterexample or using a proof by contradiction.

5. What are some real-world applications of normed vector spaces?

Normed vector spaces have many real-world applications, such as in physics, engineering, and computer science. They are used to model and analyze physical systems, design efficient algorithms, and perform data analysis and machine learning tasks. They are also used in economics, finance, and other fields to represent and manipulate complex data sets.

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