# Engineer stumped by pure math, need to find some way to get started

• lankman
Some common norms arethe absolute value, the square of the absolute value, and the sum of the squaresof the absolute values. Of course, in practice, one often defines Sobolev spaces and Hausdorff spaces and soforth, depending on what sorts of functions are being considered. In summary, a functional space is a set of functions having some particular property, where some "topology" is defined. A scalar product of two functions in a functional space is a member-function, and a Hilbert space is a special type of functional space with some very nice properties.

#### lankman

Hi guys, I am reading some paper on Mellin Transform and its application in signal processing and I am totally stumped by phrases such as lebesgue space, square integrable, isometry, automorphism, measurable and essential supreme and such.

Of course, when I look them up one by one I sort of get the concept (although still very confused about why people would go through all that mess to define these concepts as such), but I would like to be comprehensive about it.

Where can I get a systematic understanding of these terms? Do I need a course in real analysis? If possible, can someone elaborate as to what a functional space mean?

Some of those words do imply real analysis.

A functional space is a set of functions having some particular property, where some "topology" is defined. Which usually means there is a way to measure how "far apart" two functions are. Very frequently functional spaces are linear, which means that for any two numbers a and b, and any two member-functions f and g, af + bg is also a member-function. In those space you typically have a scalar product (f, g), which is used to define the norm ||f|| = |(f, f)|, and the norm is used to define the metric r(f, g) = ||f - g||, which tells you how far apart f and g are.

Taking the space of square integrable functions on the interval [a, b], the scalar product of f and g is simply the definite integral of the product f(t)g(t) over the interval [a, b]. What is "square integrable"? This a function integral of whose square is finite, which simply means its norm is finite, so all the definitions are self-consistent.

lankman said:
Hi guys, I am reading some paper on Mellin Transform and its application in signal processing and I am totally stumped by phrases such as lebesgue space, square integrable, isometry, automorphism, measurable and essential supreme and such.

Of course, when I look them up one by one I sort of get the concept (although still very confused about why people would go through all that mess to define these concepts as such), but I would like to be comprehensive about it.

Where can I get a systematic understanding of these terms? Do I need a course in real analysis? If possible, can someone elaborate as to what a functional space mean?

One of the important things about square-integrability is that the space of square-

integrable functions, laid out as voko did, is a Hilbert space, and Hilbert spaces have

some very nice properties. To add a bit about norms, there are often many

different ones used ( and inequivalent topologically), reflecting the different notions

of what functions being closed to each other may mean.