Ficken's characterization of inner product spaces.

Question

Ficken Characterization : Given X is normed space. For any vectors x,y in X and if $\Vert x \Vert = \Vert y\Vert$, then $\Vert ax+by\Vert = \Vert bx + ay \Vert$ , for any scalars a & b. The conclusion: X is inner product space. Ficken modify $\Vert x+y \Vert = \Vert x-y+2y \Vert$ and $\Vert y-x+2x \Vert$, $\Vert x+y-2y\Vert$, $\Vert x+y-2x \Vert$.

On page 364, Ficken obtains 4 equation:

1. $\Vert y \Vert \Vert x+y\Vert\Vert x-y \Vert = \big\Vert 2 \Vert y \Vert^2x+(\Vert x-y \Vert^2 - 2 \Vert y \Vert^2)y \big \Vert$
2. $\Vert y\Vert \Vert x+y\Vert \Vert x-y\Vert = \big \Vert 2\Vert y\Vert ^2x + (\Vert x-y\Vert^2 - 2\Vert x \Vert^2)y \big \Vert$
3. $\Vert y\Vert \Vert x+y\Vert \Vert x-y\Vert = \big \Vert 2\Vert y\Vert ^2x + (2\Vert y \Vert^2 -\Vert x+y\Vert^2)y \big \Vert$
4. $\Vert y\Vert \Vert x+y\Vert \Vert x-y\Vert = \big \Vert 2\Vert y\Vert ^2x + (2\Vert x \Vert^2 -\Vert x+y\Vert^2)y \big \Vert$

for all x,y.

Especially on page 364 second and fourth equation, I don't know how to get it. Can somebody help me? Thanks in advanced.

PS: You can check a full paper version in here https://sci-hub.tw/10.2307/1969273.

On the first equation, we can modify $\Vert x+y \Vert = \Vert x-y + 2y \Vert = \big\Vert (x-y) + 2\frac{\Vert y \Vert \Vert x-y\Vert}{\Vert y \Vert \Vert x-y \Vert} y \big\Vert$. Because $\Vert x-y \Vert = \big\Vert \frac{\Vert x-y \Vert}{\Vert y \Vert} y \big\Vert$, then we can do, $\big \Vert \frac{2\Vert y \Vert}{\Vert x-y \Vert}(x-y) + \frac{\Vert x - y \Vert}{\Vert y \Vert}y \big \Vert$. By algebraic manipulation we get $\Vert x + y \Vert = \frac{\Vert 2 \Vert y \Vert^2 x + (\Vert x-y \Vert^2 - 2 \Vert y \Vert^2) y \Vert}{\Vert y \Vert \Vert x-y \Vert}$.

On the third equation can be solved with same trick. But I still confuse how to get second and fourth equation. I want to show second equation and fourth equation, for all x,y.

Answer 1

I have already found the way.

On the second equation we can write: $\Vert x+y \Vert = \Vert (y-x) + 2x \Vert = \Vert (y-x) + 2\frac{\Vert y-x\Vert \Vert x \Vert}{\Vert y-x\Vert \Vert x \Vert}x \Vert$. With the same trick like the first equation, we get $\Vert x \Vert \Vert x+y \Vert \Vert x-y \Vert = \Big\Vert 2 \Vert x \Vert^2y + (\Vert x-y\Vert^2 - 2\Vert x \Vert^2)x \Big\Vert.$ Multiply with $\frac{\Vert y \Vert}{\Vert x \Vert}$, and we obtain $\Vert y \Vert \Vert x+y \Vert \Vert x-y \Vert = \Big\Vert 2 \Vert y \Vert \Vert x \Vert y + (\Vert x-y\Vert^2 - 2\Vert x \Vert^2)x \frac{\Vert y \Vert}{\Vert x \Vert}\Big\Vert.$

Because $\Vert y \Vert = x \frac{\Vert y \Vert}{\Vert x \Vert}$, we can switch the constant. Finally $ \Vert y \Vert\Vert x+y \Vert \Vert x-y \Vert = \Big\Vert 2 \Vert y \Vert\Vert x \Vert \frac{\Vert y \Vert}{\Vert x \Vert}x+ (\Vert x-y \Vert^2 - 2 \Vert x \Vert^2) y \Big\Vert.$ $ \Vert y \Vert\Vert x+y \Vert \Vert x-y \Vert = \Big\Vert 2 \Vert y \Vert^2 x + (\Vert x-y \Vert^2 - 2 \Vert x \Vert^2) y \Big\Vert.$

And the second equation is proved. With the same trick, we can obtain the fourth equation. Thank you.

Answer 2

Your link does not work for me. However, I found a paper

À la recherche de la preuve perdue: a simple proof of the Ficken theorem by Dariusz Cichoń,

in which it is stated:

The proof given in [7] is elementary, almost four page long and incorrect. There is no mistake, but a flaw in the reasoning: one cannot see any proper way of obtaining similar conditions on page 364.

Reference [7] is the paper by Ficken, and the problematic page referenced is the one you are struggling with. So it appears that there is no elementary way to get these inequalities and you should not be tearing your hair out over them.

The above paper presents an elementary proof of the theorem. Google searching yields also a PhD thesis entitled "Characterizations of Inner Product Spaces" by John Arthur Oman in which the theorem is proven twice (and supposedly is a corollary in a third result) as well as a book by D. Amir (with exactly the same title) containing a proof of the result.