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I have been looking around for this question, but all results I found only describe the definition and not the answer I seek.

Is "kernel" basically a synonym of "function"? When should be the time we should use the word "kernel" instead of "function"?

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"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.

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Is it old-fashioned? What would you name the Fejér kernel then in modern language? It looks to me like it is still in use in harmonic analysis. –  Jonas Teuwen Feb 15 '11 at 9:19
    
@Jonas: "old-fashioned" doesn't mean "out of use." Perhaps that was bad choice of words. –  Qiaochu Yuan Feb 15 '11 at 9:34
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A kernel is not a synonym for a function. As Zhen Lin said, it is a property of of a function.

I will try to explain the kernel of a map in linear algebra. Suppose we have a linear map $L:V\rightarrow W$. Then the kernel of $L$ consists of all vectors in $V$ which are mapped to zero. $0$ is always mapped to zero, thus the kernel is never empty. It is easy to prove that the kernel of $L$ is always a linear subspace of $V$. If $v,w\in V$ are vectors in the kernel of $L$, that is $Lv=Lw=0$, then $L(\lambda v+\mu w)=\lambda Lv+\mu L w=\lambda 0+\mu 0=0$. Thus any linear combination of vectors in the kernel is an element of the kernel. Hence it is a linear subspace of $V$.

This idea generalizes to a lot of other mathematical constructions (e.g. groups), this was what Zhen Lin was talking about.

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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.

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I'm not sure if I understand what you said... –  Karl Feb 15 '11 at 7:57
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I don't think this is what the OP means. –  Qiaochu Yuan Feb 15 '11 at 9:00
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They are not the same. A kernel is a set of elements in the domain whereas a function is a mapping from the domain to the codomain.

Compare kernel and function.

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The word "kernel" in mathematics has several different meanings. Zhen Lin addressed one of them - and maybe not the one you are referring too. So maybe you should specify which area of mathematics you are considering in your question.

H

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Hagen, this should really be a comment. –  Qiaochu Yuan Feb 15 '11 at 9:12
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