Defining Real-Valued Scalar Product in Vector Spaces

In summary, the Jordan–von Neumann theorem states that in a vector space over a field of real numbers, a positive defined norm that satisfies the triangle inequality and the equation ||aU||=|a|*||u|| can define a real valued scalar product. This scalar product is defined as (U,V)=1/2*(||U+V||^2 - ||U||^2- ||V||^2), and it satisfies the postulates of symmetry, linearity in the first argument, and positive-definiteness, if the identity ||U+V||^2 + ||U-V||^2 = 2*||U||^2 + 2*||V||^2 is satisfied by the
  • #1
FilipVz
8
0
Hi,

can somebody help me with the problem:

Suppose that in a vector space over field of real numbers a positive defined norm is defined for each vector which satisfies the triangle inequality and ||aU||=|a|*||u||. Show that a real valued scalar product can de defined as follows:

(U,V)=1/2*(||U+V||^2 - ||U||^2- ||V||^2)

which satisfied the postulates, if following identity is satisfied by the norms:

||U+V||^2 + ||U-V||^2 = 2*||U||^2 + 2*||V||^2
 
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  • #2
FilipVz said:
Hi,

can somebody help me with the problem:

Suppose that in a vector space over field of real numbers a positive defined norm is defined for each vector which satisfies the triangle inequality and ||aU||=|a|*||u||. Show that a real valued scalar product can de defined as follows:

(U,V)=1/2*(||U+V||^2 - ||U||^2- ||V||^2)

which satisfied the postulates, if following identity is satisfied by the norms:

||U+V||^2 + ||U-V||^2 = 2*||U||^2 + 2*||V||^2

Welcome to MHB, FilipVz! :)

How far do you get with the postulates?

That is:
  1. (Conjugate) symmetry
  2. Linearity in the first argument
  3. Positive definiteness
See e.g. wiki.

For starters, is it symmetric?
That is, does for all u and v hold that (u,v) = (v,u)?
 
  • #3
I like Serena said:
Welcome to MHB, FilipVz! :)

How far do you get with the postulates?

That is:
  1. (Conjugate) symmetry
  2. Linearity in the first argument
  3. Positive definiteness
See e.g. wiki.

For starters, is it symmetric?
That is, does for all u and v hold that (u,v) = (v,u)?
Hi, I like Serena,

Scalar product is symmetric.

Could you please explain to me what is "Linearity in the first argument"?

Thanks,

Filip
 
  • #4
FilipVz said:
Hi, I like Serena,

Scalar product is symmetric.

Could you please explain to me what is "Linearity in the first argument"?

Thanks,

Filip

From the wiki reference I gave you can see that an inner product over the field of the real numbers must satisfy the following axioms:

1. Symmetry:
$$\langle x,y\rangle =\langle y,x\rangle.$$
2. Linearity in the first argument:
$$\langle ax,y\rangle= a \langle x,y\rangle.$$
$$\langle x+y,z\rangle= \langle x,z\rangle+ \langle y,z\rangle.$$
3. Positive-definiteness:
$$\langle x,x\rangle \geq 0 \text{ with equality only for }x = 0.$$Yes, your inner product is symmetric, but you did not mention why.
The reason is in the expression for your inner product you can swap u and v around, and you'll end up with the same expression.Next step is linearity for a factor in the first argument.
To find if it is, you need to write your inner product with $a \mathbf u$ and $\mathbf v$ instead of $\mathbf u$ and $\mathbf v$.
And then figure out if the resulting expression is the same as it was for $\mathbf u$ and $\mathbf v$, except for a factor $a$.
 
  • #5
I like Serena said:
From the wiki reference I gave you can see that an inner product over the field of the real numbers must satisfy the following axioms:

1. Symmetry:
$$\langle x,y\rangle =\langle y,x\rangle.$$
2. Linearity in the first argument:
$$\langle ax,y\rangle= a \langle x,y\rangle.$$
$$\langle x+y,z\rangle= \langle x,z\rangle+ \langle y,z\rangle.$$
3. Positive-definiteness:
$$\langle x,x\rangle \geq 0 \text{ with equality only for }x = 0.$$Yes, your inner product is symmetric, but you did not mention why.
The reason is in the expression for your inner product you can swap u and v around, and you'll end up with the same expression.Next step is linearity for a factor in the first argument.
To find if it is, you need to write your inner product with $a \mathbf u$ and $\mathbf v$ instead of $\mathbf u$ and $\mathbf v$.
And then figure out if the resulting expression is the same as it was for $\mathbf u$ and $\mathbf v$, except for a factor $a$.

Linearity for a factor in the firs arugment: (aU,V)=a(U,V)

Positive-definiteness: (U,U)>=0
(U,U)=0, iff U=0

What is the next step?
 
  • #6
FilipVz said:
Linearity for a factor in the firs arugment: (aU,V)=a(U,V)

Positive-definiteness: (U,U)>=0
(U,U)=0, iff U=0

What is the next step?

You skipped the step for linearity of addition in the first argument.

After that, there is no next step.
If you can prove that they are satisfied, you are done - then it is an inner product.
You didn't really verify or prove them yet though.
 
  • #7
I like Serena said:
You skipped the step for linearity of addition in the first argument.

After that, there is no next step.
If you can prove that they are satisfied, you are done - then it is an inner product.
You didn't really verify or prove them yet though.

So, all i need to do is to prove the postulates of Scalar product?
 
  • #8
FilipVz said:
Hi,

can somebody help me with the problem:

Suppose that in a vector space over field of real numbers a positive defined norm is defined for each vector which satisfies the triangle inequality and ||aU||=|a|*||u||. Show that a real valued scalar product can de defined as follows:

(U,V)=1/2*(||U+V||^2 - ||U||^2- ||V||^2)

which satisfied the postulates, if following identity is satisfied by the norms:

||U+V||^2 + ||U-V||^2 = 2*||U||^2 + 2*||V||^2
This is the Jordan–von Neumann theorem. As I like Serena has pointed out, you need to show that the inner product satisfies the additivity property $\langle x+y,z\rangle = \langle x,z\rangle + \langle y,z\rangle$ and the scalar multiplication property $\langle ax,y\rangle = a\langle x,y\rangle$. Neither of those is at all easy. Unless you are a budding Jordan or von Neumann, you are unlikely to be able to find a proof unaided.

The proof is usually given (as in the above link) for a vector space over the complex numbers. The proof for the real case follows the same route and is basically no easier.
 

Related to Defining Real-Valued Scalar Product in Vector Spaces

1. What is a real-valued scalar product in vector spaces?

A real-valued scalar product in vector spaces is a mathematical operation that takes two vectors as inputs and produces a real number as an output. It is a way of measuring the similarity or angle between two vectors in a multi-dimensional space.

2. What are the properties of a real-valued scalar product?

The properties of a real-valued scalar product include commutativity, distributivity, and additivity. This means that the order of the vectors does not matter, the product can be distributed over addition of vectors, and the product of a vector and a sum of vectors is equal to the sum of the products of the vector with each individual vector.

3. How is a real-valued scalar product different from a complex-valued scalar product?

A real-valued scalar product only produces real numbers as outputs, whereas a complex-valued scalar product can produce complex numbers as outputs. Additionally, a complex-valued scalar product is not commutative, meaning the order of the vectors does matter.

4. What is the significance of a real-valued scalar product in vector spaces?

A real-valued scalar product is significant because it allows us to define the concept of perpendicularity or orthogonality in vector spaces. It is also useful in many applications, such as in physics, engineering, and computer graphics.

5. How is a real-valued scalar product calculated?

A real-valued scalar product is calculated by taking the dot product of two vectors. This involves multiplying the corresponding components of the two vectors and then adding the products together. The result is a single real number representing the similarity or angle between the two vectors.

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