MHB Defining Real-Valued Scalar Product in Vector Spaces

[FilipVz]

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

 

[I like Serena]

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)?

 

[FilipVz]

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

 

[I like Serena]

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 $a$.

 

[FilipVz]

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?

 

[I like Serena]

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.

 

[FilipVz]

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?

 

[Opalg]

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.