What is the difference between a kernel and a function?

Solution 1:

"Kernel" is an old-fashioned term for the function you use to define certain integral operators. (I assume this is the sense you mean, not the more common modern sense, which is completely different.) Like many other words in mathematics (although people generally never tell you this), it has less to do with denotation than connotation: when you use the word "kernel" you are thinking of your function in terms of integral operators.

Solution 2:

They aren't synonymous. A kernel is a property of a function. Most generically, if you have a function $f: X \to Y$, it is defined as the equivalence relation on X which identifies $x_1$ and $x_2$ if and only if $f(x_1) = f(x_2)$.

There are specialisations of this depending on the category $f$ lives in: for example, if $f$ is a group homomorphism, the homogeneity of the group structure means that we can represent this equivalence relation as the normal subgroup $\{ x \in X : f(x) = e \}$. Similarly for $f$ a linear map of vector spaces, or modules, or rings.