Normed linear space and inner product space

In summary, the conversation discusses the difference between normed linear spaces and inner product spaces, and the task of finding examples of normed linear spaces that are not inner product spaces. The suggestion is made to look for properties satisfied by inner product spaces, such as positive definiteness and the parallelogram law. The conversation ends with the suggestion to work with a specific example, such as R^2, to find different norms and their associated inner products.
  • #1
cabin5
18
0

Homework Statement


Not all normed linear spaces are inner product spaces. Give examples.


Homework Equations


all equations and conditions constructing inner and normed linear spaces.


The Attempt at a Solution


Well, I tried some of spaces like L space, but I didn't find any logical solutions. Frankly, I don't know from where to begin to prove this.
 
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  • #2
Try to flip through your book to find a property inner product spaces satisfy that is stated in terms of the norm generated by the inner product.

If you want another hint, post back.
 
  • #3
I know all the sufficient properties for an inner product space, but how can I find a suitable example for this particular problem?
 
  • #4
Did you do what I suggested? What sufficient properties did you find?
 
  • #5
ah, do you mean positive definiteness or parallelogram law?
 
  • #6
The latter is very useful!
 
  • #7
yes,ok, but how should I write down the associated norm of a space which I don't know.

the only thing that I can write down is the parallelogram law itself which automatically yields the same thing in terms of inner product of its associated norm.

I am a bit confused...
 
  • #8
Work with a specific example. Take R^2 for instance. What norms do you know on this space?
 
  • #9
thanks for the post!
 

1. What is the difference between a normed linear space and an inner product space?

A normed linear space is a vector space equipped with a norm function, which measures the length or magnitude of a vector. An inner product space is a vector space equipped with an inner product, which is a generalization of the dot product that also takes into account the direction of the vectors. In other words, an inner product space not only measures the length of a vector, but also the angle between two vectors.

2. How do you define a norm in a normed linear space?

A norm in a normed linear space is a function that assigns a non-negative value to each vector in the space, with the properties of being positive definite, homogeneous, and satisfying the triangle inequality. In other words, the norm of a vector is always greater than or equal to zero, multiplying a vector by a scalar multiplies its norm by the absolute value of that scalar, and the norm of a sum of vectors is always less than or equal to the sum of their individual norms.

3. What is the significance of the inner product in an inner product space?

The inner product in an inner product space provides a way to measure the similarity or orthogonality of two vectors. This is useful in a variety of applications, such as in quantum mechanics where the inner product is used to calculate probabilities of outcomes, or in signal processing where the inner product is used to determine how similar or different two signals are.

4. Can an inner product space also be a normed linear space?

Yes, an inner product space can also be a normed linear space. This is because the inner product defines a norm in the space, where the norm of a vector is the square root of the inner product of the vector with itself. Therefore, any inner product space also has a defined norm, making it a normed linear space as well.

5. How are normed linear spaces and inner product spaces used in real-world applications?

Normed linear spaces and inner product spaces are used in a wide range of scientific and engineering fields, including physics, engineering, and computer science. They are used to model physical systems, analyze data, and develop algorithms for solving complex problems. For example, inner product spaces are used in computer vision to measure the similarity between images, and normed linear spaces are used in optimization problems to find the minimum or maximum of a function.

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