Confusione sul modulo quadro di una funzione

[javicemarpe]

The problem is that the integral formula you wrote is not the square of the modulus but the square of the $L^2$-norm of the function $\phi$.

Ok...so the "square of the modulus" is $\bar\phi(x)\phi(x)$ while the square of the $L^2$-norm is $\int_a^b|\phi(x)|^2dx$ ...? but why are they both indicated as $|\phi|^2$ if they are different objects?

Actually, I've never seen that notation for the $L^2$-norm. It is usually denoted by $||\phi||_{L^2}$. Sometimes people doesn't choose the most convenient notation.

[Giud]

ecco

a norma dipende da come è stato definito il prodotto, giusto?

certo. fissato un prodotto scalare definito positivo la norma al quadrato è scrivibile come il prodotto scalare di un vettore per sè stesso. negli spazi di hilbert questa norma soddisfa poi la regola del parallelogramma.
il modulo è il classico modulo dell'analisi complessa.

Grazie mille!

[Giud]

The problem is that the integral formula you wrote is not the square of the modulus but the square of the $L^2$-norm of the function $\phi$.

Ok...so the "square of the modulus" is $\bar\phi(x)\phi(x)$ while the square of the $L^2$-norm is $\int_a^b|\phi(x)|^2dx$ ...? but why are they both indicated as $|\phi|^2$ if they are different objects?

Actually, I've never seen that notation for the $L^2$-norm. It is usually denoted by $||\phi||_{L^2}$. Sometimes people doesn't choose the most convenient notation.

Yes! Please take a look at this: http://mathworld.wolfram.com/L2-Norm.html

[javicemarpe]

Actually, I've never seen that notation for the $L^2$-norm. It is usually denoted by $||\phi||_{L^2}$. Sometimes people doesn't choose the most convenient notation.

Yes! Please take a look at this: http://mathworld.wolfram.com/L2-Norm.html

It's just a wrong choice of notation, because it produces this confusion. I would write (as I said before) $||\phi||_{L^2}$ for the $L^2$-norm. This is the most common notation for this norm and it doesn't lead you to any confusion.

On the other hand, in my opinion, this notation can be justified:

When you read $|\phi|$ (without talking about any point $x$) it should be understood that it is an object which measures in some sense the "size" of the function $\phi$ as a function, namely the $L^2$-norm (there exist more possible ways to measure "how big" $\phi$ is, for instance, any $L^p$-norm).

If you read $|\phi(x)|$, then it should be understood that what we are measuring is the "size" of the complex number $\phi(x)$, and a canonical way to do this is by using its complex modulus ($|z|=z\overline{z}$).

[Giud]

Actually, I've never seen that notation for the $L^2$-norm. It is usually denoted by $||\phi||_{L^2}$. Sometimes people doesn't choose the most convenient notation.

Yes! Please take a look at this: http://mathworld.wolfram.com/L2-Norm.html

It's just a wrong choice of notation, because it produces this confusion. I would write (as I said before) $||\phi||_{L^2}$ for the $L^2$-norm. This is the most common notation for this norm and it doesn't lead you to any confusion.

On the other hand, in my opinion, this notation can be justified:

When you read $|\phi|$ (without talking about any point $x$) it should be understood that it is an object which measures in some sense the "size" of the function $\phi$ as a function, namely the $L^2$-norm (there exist more possible ways to measure "how big" $\phi$ is, for instance, any $L^p$-norm).

If you read $|\phi(x)|$, then it should be understood that what we are measuring is the "size" of the complex number $\phi(x)$, and a canonical way to do this is by using its complex modulus ($|z|=z\overline{z}$).

This really helps! Thanks!!
Thank you both