Showing a Norm is not an Inner Product

In summary, the taxicab norm, given by ||v||=|x_{1}|+...+|x_{n}|, is not an inner product. This can be easily shown by examining the parallelogram law, which leads to a contradiction when applied to the taxicab norm. This norm does not satisfy linearity, but it does satisfy the triangle inequality. Therefore, ||v+w|| is not an inner product.
  • #1
Punkyc7
420
0
Show the taxicab norm is not an IP.

taxicab norm is v=(x[itex]_{1}[/itex]...x[itex]_{n}[/itex])
then ||V||= |x[itex]_{1}[/itex]|+...+|x[itex]_{n}[/itex]|)

I was thinking about using the parallelogram law

but I would get this nasty thing(|x[itex]_{1}[/itex]+w[itex]_{1}[/itex]|+...+|x[itex]_{n}[/itex]+w[itex]_{n}[/itex]|)[itex]^{2}[/itex]+(|x[itex]_{1}[/itex]-w[itex]_{1}[/itex]}+...+|x[itex]_{n}[/itex]-w[itex]_{n}[/itex]|[itex])^{2}[/itex]=2(|x[itex]_{1}[/itex]|+...+|x[itex]_{n}[/itex]|)[itex]^{2}[/itex]+(|w[itex]_{1}[/itex]|+...+|w[itex]_{n}[/itex]|)[itex]^{2}[/itex]Im not sure how to work with this. Am I going about this wrong?

Also there might be some typos with the absolute value signs the latex get messy
 
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  • #2
Hint: the norm doesn't satisfy linearity (it does satisfy the triangle inequality).
 
  • #3
Do you mean that ||v+w||is not an inner product? This is easy to show! Just go through the definition of inner product -- one of the axioms is clearly not satisfied in this case.
 

FAQ: Showing a Norm is not an Inner Product

1. What is a norm?

A norm is a mathematical concept used to measure the size or length of a vector in a vector space. It is a function that assigns a non-negative value to a vector, and the value is equal to 0 if and only if the vector is a zero vector.

2. How is a norm different from an inner product?

A norm and an inner product are both mathematical concepts used in vector spaces, but they serve different purposes. A norm measures the size or length of a vector, while an inner product measures the angle or distance between two vectors. Additionally, an inner product is a function that takes in two vectors and outputs a scalar, while a norm outputs a non-negative value.

3. Can a norm also be an inner product?

Yes, in certain cases, a norm can also be an inner product. For example, in a real vector space with the Euclidean norm, the dot product serves as both a norm and an inner product. However, not all norms can be inner products, and not all inner products can be norms.

4. How can you show that a norm is not an inner product?

To show that a norm is not an inner product, we can look at the properties of an inner product and see if they hold for the given norm. If any of the properties do not hold, then we can conclude that the norm is not an inner product. For example, if the norm does not satisfy the symmetry property, where the inner product of two vectors is equal to the inner product of the vectors in reverse order, then it is not an inner product.

5. Why is it important to show that a norm is not an inner product?

It is important to show that a norm is not an inner product because it helps us understand the properties and limitations of different mathematical concepts. Additionally, it allows us to determine which operations can and cannot be performed on the given vector space. This can be useful in various applications, such as computer graphics, signal processing, and optimization problems.

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