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What is a proper kernel
Definition: A finitely positive semi-definite function
is a symmetric function of its arguments for which matrices formed
by restriction on any finite subset of points is positive semi-definite.
k : x× y→ R
α TKα ≥ 0 ∀α
Theorem: A function can be written
as where is a feature map
iff k(x,y) satisfies the semi-definiteness property.
k : x× y→ R
k(x, y) =< Φ(x),Φ( y) > Φ(x)
x→Φ(x)∈F
Relevance: We can now check if k(x,y) is a proper kernel using
only properties of k(x,y) itself,
i.e. without the need to know the feature map!

有道理