Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I try to understand operator-valued kernels. For this purpose, first want to know what is an operator. I can see the definition of operator here, but I do not quit get it. Can anyone explain it in simple words, maybe with examples?

share|improve this question
add comment

1 Answer

up vote 6 down vote accepted

An operator is a special kind of function. The simplest functions take a number as an input and give a number as an output. Operators take a function as an input and give a function as an output.

As an example, consider $\Omega$, an operator on the set of functions $\mathbb{R} \to \mathbb{R}.$ We can define $\Omega(f) := f + 1$. The operator $\Omega$ takes the function $x \mapsto f(x)$ as an input and gives $x \mapsto f(x)+1$ as its output function.

Another, well known, linear operator is differentiation. In this example:

$$\Omega(f) := \frac{df}{dx} \, . $$

It is a linear operator because $\Omega(\lambda f+\mu g) = \lambda\Omega(f) + \mu\Omega(g).$

Functions are increadibly general objects, so operators are even more so. Operators are functions on functions. If you're still stuck then I recommend you spend more time thinking about functions.

share|improve this answer
Thanks a lot. Also, can can an operator be applied to many arguments? Like $\Omega(f, \, g) = f+g $? –  user25004 Sep 29 '12 at 1:50
Next, what does "If the range is on the real line or in the complex plane, the mapping is usually called a functional instead" mean? Because, by the explanations above the range should be a set of functions, not numbers. –  user25004 Sep 29 '12 at 1:57
@user25004 In the case of a functional, the input is a function while the output is a number. You can think of the numbers as being the constant functions, i.e. $f(x) = c$ for all $x$. –  Fly by Night Sep 29 '12 at 15:01
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.