10 Lemmas about LP Norm
For a function \(f\) with domain \(A\)
\[ \lVert f \rVert _{\ell ^1} \leq \sqrt{|A|} \lVert f \rVert _{\ell ^2} \]
Proof
This is a particular case of the Cauchy-Schwartz inequality.
For a function \(f\) with domain \(A\)
\[ \lVert f \rVert _{\ell ^p} = |A|^{1/p} \lVert f \rVert _{L^p} \]
Proof
Trivial from the definition of \(\lVert \cdot \rVert _{L^p}\).
\[ \lVert f \rVert _{\ell ^p} \leq |A|^{1/p} \lVert f \rVert _{\ell ^\infty } \]
Proof
\[ (\sum _{x} |f(x)|^p)^{1/p} \leq (\sum _{x} \lVert f \rVert _{\ell ^\infty }^p)^{1/p} = (|A| \lVert f \rVert _{\ell ^\infty }^p)^{1/p} = |A|^{1/p} \lVert f \rVert _{\ell ^\infty } \]
.
Note that in this lemma \(\langle f, g \rangle \) is \(\sum _x \bar{f(x)} {g(x)}\).
\[ |\langle f, g \rangle | \leq \lVert f \rVert _{\ell ^1} \lVert g \rVert _{\ell ^\infty } \]
Proof
\[ |\sum _x \bar{f(x)} {g(x)}| \leq \sum _x |\bar{f(x)} {g(x)}| \leq \sum _x |f(x)| \lVert g \rVert _{\ell ^\infty } = \lVert f \rVert _{\ell ^1} \lVert g \rVert _{\ell ^\infty } \]
\[ \lVert a \rVert _{\ell ^2} \leq \sqrt{\lVert a \rVert _{\ell ^1} \lVert a \rVert _{\ell ^\infty }} \]
Proof
Trivial with 10.4 and \(\lVert a \rVert _{\ell ^2} = \sqrt{\langle a, a \rangle }\)