important of Hilbert space

[AntonioLao]

In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm (distance) defined by the inner product. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion and certain linear transformations such as the Fourier transform. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics, although many basic features of quantum mechanics can be understood without going into details about Hilbert spaces. Hilbert spaces are studied in the branch of math called functional analysis.

[Guille]

What is the norm of inner product?

What other types o flinear transformations are there apart from fourier trasform?

What other things are studied in functional analyse apart from Hilbert spaces?

[AntonioLao]

What is the norm of inner product?

It is another way of saying the validity of Pythagorean theorem for finding the distance between two points.

What other types o flinear transformations are there apart from fourier trasform?

Eigenvalue solutions in solving partial differential equations. The existence of diagonalizable Hermitian matrices.

What other things are studied in functional analyse apart from Hilbert spaces?

Functions. All kinds of functions.

[Guille]

Is there any mathematical branche that studies the relationships between matrices, tranformations and functions?

[AntonioLao]

One of the candidates is linear algebra of linear functional operators.

[Guille]

One of the candidates is linear algebra of linear functional operators.

Why do transformation always have to do with linear _____? Why can't it deal with non-linear _______?

[AntonioLao]

Why do transformation always have to do with linear? Why can't it deal with non-linear?

Not always. The transformations dilation, reflection and inversion are nonlinear. Linear transformations are easier to work with.

[Guille]

Not always. The transformations dilation, reflection and inversion are nonlinear. Linear transformations are easier to work with.

Out of the three the only one I don't automatically recognise/visualize is dilation, what is it?

[AntonioLao]

dilation, what is it?

Change in size or scale factor. There are practical engineering designs that cannot work by scaling. For example, increasing the the diameter of a suspension cable does not necessarily increasing its tensile strength (however, increasing the number of cables in bundle does). In special relativity, dilation is used for slowing of time with respect to frame motion approaching light speed.

[hanzoganz]

Very interesting post. I really like functional analysis because it has such a powerfull tools. Quantum physiscs is based on Hilbert Spaces, so, as you say, we can understand Heisenbergs uncertainty principle by a Fourier transform on a Hilbert space. I work with stability of nonlinear dynamical systems and yes, linearity is much more easier to work with and gives us enough hints about the nonlinear behaviour.
We could say that functional anaysis deals with functionals and operators. the first are mapping from a space into R. the second are mapping from one space to other. I think this field of math uses a lot of imagination. i invite you to read and work a bit on this.

cheers