convergence in L-p implies convergence in cut norm

[[concept]]

Theorem

Let $W:[0,1{]}^2\to [-1,1]$. Then $||W|{|}_{\square }\stackrel{(1)}{\le }||W|{|}_1\stackrel{(2)}{\le }||W|{|}_2\stackrel{(3)}{\le }||W|{|}_{\infty }\stackrel{(4)}{\le }1$ And convergence $L_p$ for $p\ge 1$ implies convergence in the cut norm

Proof

Easy to see (1) from the definition of the cut norm:

cut norm (kernels) $W$ be a kernel in $[0,1{]}^2$. Its cut norm is defined as $\lvert \lvert W \rvert \rvert_{\square} = \sup_{S,T \subseteq [0,1]} \left|\int \int _{S\times T} W(u,v), du , dv \right|$$

Let

This is computing the L-1 norm restricted to a subset of the nodes, so $||W|{|}_{\square }\le ||W|{|}_1$.

(2) and (3) are a common result from functional analysis, and (4) is because of the selected codomain for $W$.

Convergence in the cut norm follows immediately from this hierarchy.

Mentions

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