Abstract¶ The conditions for a functional
Keywords: Norms Vector Spaces Vectors Matrices ¶ Norm Conditions ¶ Let V V V F \mathbb{F} F ∣ ∣ ⋅ ∣ ∣ \vert \vert \cdot \vert \vert ∣∣ ⋅ ∣∣ V V V ∣ ∣ ⋅ ∣ ∣ : V → R + \vert \vert \cdot \vert \vert: V \rightarrow \mathbb{R}^{+} ∣∣ ⋅ ∣∣ : V → R + ψ , σ ∈ V \psi, \sigma ~\in ~V ψ , σ ∈ V λ ∈ F 1 \lambda~\in~\mathbb{F}^{1} λ ∈ F 1
∣ ∣ ψ ∣ ∣ ≥ 0 \vert \vert \psi \vert \vert \geq 0 ∣∣ ψ ∣∣ ≥ 0 ∣ ∣ ψ ∣ ∣ = 0 \vert \vert \psi \vert \vert=0 ∣∣ ψ ∣∣ = 0 ψ = 0 \psi = 0 ψ = 0
∣ ∣ λ ψ ∣ ∣ = ∣ λ ∣ ∣ ∣ ψ ∣ ∣ \vert \vert \lambda \psi \vert \vert = \vert \lambda \vert ~ \vert \vert \psi \vert \vert ∣∣ λ ψ ∣∣ = ∣ λ ∣ ∣∣ ψ ∣∣
∣ ∣ ψ + σ ∣ ∣ ≤ ∣ ∣ ψ ∣ ∣ + ∣ ∣ σ ∣ ∣ \vert \vert \psi + \sigma \vert \vert \leq \vert \vert \psi \vert \vert + \vert \vert \sigma \vert \vert ∣∣ ψ + σ ∣∣ ≤ ∣∣ ψ ∣∣ + ∣∣ σ ∣∣
Norms can be thought of as functionals that measure the size of an elements of a vector space. A vector space with a norm is called a normed vector space.
Vector Norm Examples ¶
Let ψ ∈ V \psi~\in~V ψ ∈ V ψ ∈ F n \psi~\in~\mathbb{F}^{n} ψ ∈ F n ( ψ 1 , ψ 2 , … , ψ n ) (\psi_1, \psi_2, \ldots, \psi_n) ( ψ 1 , ψ 2 , … , ψ n ) l p l_{p} l p
∣ ∣ ψ ∣ ∣ p = ( ∣ ψ 1 ∣ p + ∣ ψ 2 ∣ p + … + ∣ ψ n ∣ p ) 1 / p \vert \vert \psi \vert \vert_{p} = \big( \vert \psi_1 \vert^p + \vert \psi_2 \vert^p + \ldots + \vert \psi_n \vert^p)^{1/p} ∣∣ ψ ∣ ∣ p = ( ∣ ψ 1 ∣ p + ∣ ψ 2 ∣ p + … + ∣ ψ n ∣ p ) 1/ p With p = 1 p=1 p = 1 p = 2 p=2 p = 2 l 2 l_2 l 2
∣ ∣ ψ ∣ ∣ 1 = ( ∣ ψ 1 ∣ + ∣ ψ 2 ∣ + … + ∣ ψ n ∣ ) ∣ ∣ ψ ∣ ∣ 2 = ( ∣ ψ 1 ∣ 2 + ∣ ψ 2 ∣ 2 + … + ∣ ψ n ∣ 2 ) 1 / 2 \begin{align*}
\vert \vert \psi \vert \vert_{1} &= \big( \vert \psi_1 \vert + \vert \psi_2 \vert + \ldots + \vert \psi_n \vert) \\
\vert \vert \psi \vert \vert_{2} &= \big( \vert \psi_1 \vert^2 + \vert \psi_2 \vert^2 + \ldots + \vert \psi_n \vert^2)^{1/2}
\end{align*} ∣∣ ψ ∣ ∣ 1 ∣∣ ψ ∣ ∣ 2 = ( ∣ ψ 1 ∣ + ∣ ψ 2 ∣ + … + ∣ ψ n ∣ ) = ( ∣ ψ 1 ∣ 2 + ∣ ψ 2 ∣ 2 + … + ∣ ψ n ∣ 2 ) 1/2 Let ψ ∈ V \psi~\in~V ψ ∈ V ψ ∈ F n \psi~\in~\mathbb{F}^{n} ψ ∈ F n ( ψ 1 , ψ 2 , … , ψ n ) (\psi_1, \psi_2, \ldots, \psi_n) ( ψ 1 , ψ 2 , … , ψ n )
∣ ∣ ψ ∣ ∣ ∞ = max { ∣ ψ i ∣ : 1 ≤ i ≤ n } . \vert \vert \psi \vert \vert_{\infty} = \max \{\vert \psi_i \vert~: 1 \leq i \leq n \}. ∣∣ ψ ∣ ∣ ∞ = max { ∣ ψ i ∣ : 1 ≤ i ≤ n } . This is just the maximum elements of ψ \psi ψ
Matrix Norms ¶ In addition to satisfying the above requirements, to be considered a norm on a vector space, M M M Ω : M → R + \Omega: M \rightarrow \mathbb{R}^{+} Ω : M → R + A , B ∈ M A,B~\in~M A , B ∈ M
∣ ∣ A B ∣ ∣ ≤ ∣ ∣ A ∣ ∣ ∣ ∣ B ∣ ∣ \vert \vert AB \vert \vert \leq \vert \vert A \vert \vert ~ \vert \vert B \vert \vert ∣∣ A B ∣∣ ≤ ∣∣ A ∣∣ ∣∣ B ∣∣
Matrix Norm Examples ¶ The following norms depend only on the singular values of the matrices they act upon. This means that the following conditions are true:
∣ ∣ A ∣ ∣ = ∣ ∣ A V ∣ ∣ = ∣ ∣ U A ∣ ∣ \begin{align*}
\vert \vert A \vert \vert = \vert \vert A V \vert \vert = \vert \vert U A \vert \vert
\end{align*} ∣∣ A ∣∣ = ∣∣ A V ∣∣ = ∣∣ U A ∣∣ where V V V U U U
Hence, these norms are all also unitarily invariant, meaning ∣ ∣ A ∣ ∣ = ∣ ∣ U A U † ∣ ∣ \vert \vert A \vert \vert = \vert \vert UAU^{\dagger} \vert \vert ∣∣ A ∣∣ = ∣∣ U A U † ∣∣
The singular valued decomposition of a matrix A ∈ M n m ( C ) A~\in~\mathbb{M}_{nm}(\mathbb{C}) A ∈ M nm ( C )
A = U D V † , A = U D V^{\dagger}, A = U D V † , where U U U m × m m \times m m × m V V V n × n n \times n n × n D D D m × n m \times n m × n A A A
Consider a norm ∣ ∣ ⋅ ∣ ∣ : M n m ( C ) → R + \vert \vert \cdot \vert \vert: \mathbb{M}_{nm}(\mathbb{C}) \rightarrow \mathbb{R}^+ ∣∣ ⋅ ∣∣ : M nm ( C ) → R + W W W
A W = U D V † W , = U D V ~ † , \begin{align*}
AW &= UDV^\dagger W, \\
&= UD\tilde{V}^\dagger,
\end{align*} A W = U D V † W , = U D V ~ † , where V ~ † = V † W \tilde{V}^\dagger = V^\dagger W V ~ † = V † W A A A A W AW A W D D D ∣ ∣ A W ∣ ∣ = ∣ ∣ A ∣ ∣ \vert \vert A W \vert \vert = \vert \vert A \vert \vert ∣∣ A W ∣∣ = ∣∣ A ∣∣
The same argument can be made to prove that ∣ ∣ W A ∣ ∣ = ∣ ∣ A ∣ ∣ \vert \vert W A \vert \vert = \vert \vert A \vert \vert ∣∣ W A ∣∣ = ∣∣ A ∣∣
A matrix A ∈ M n n ( C ) A \in \mathbb{M}_{nn}(\mathbb{C}) A ∈ M nn ( C ) eigen-decomposition if
A = Q P Q † \begin{align*}
A = Q P Q^\dagger
\end{align*} A = QP Q † where Q Q Q A A A P P P A A A
All matrices have a singular valued decompostion, but not all matrices have an eigen-decomposition. Moreover, unlike the singular values, the eigenvalues can be negative and complex.
For Hermitian
Note, for Hermitian
Let A ∈ M A~\in~M A ∈ M A ∈ M m n ( F ) A~\in~\mathbb{M}_{mn}(\mathbb{F}) A ∈ M mn ( F )
∣ ∣ A ∣ ∣ tr = ∑ i μ i ( A ) , \vert \vert A \vert \vert_{\textrm{tr}} = \sum_{i} \mu_{i}(A), ∣∣ A ∣ ∣ tr = i ∑ μ i ( A ) , where μ i ( A ) \mu_{i}(A) μ i ( A ) A A A A A A ∣ ∣ A ∣ ∣ 1 \vert \vert A \vert \vert_1 ∣∣ A ∣ ∣ 1
It is also given by
∣ ∣ A ∣ ∣ tr = tr ( A † A ) . \vert \vert A \vert \vert_{\textrm{tr}} = \textrm{tr} \big( \sqrt{A^{\dagger}A} \big). ∣∣ A ∣ ∣ tr = tr ( A † A ) . Let A ∈ M A~\in~M A ∈ M A ∈ M m n ( F ) A~\in~\mathbb{M}_{mn}(\mathbb{F}) A ∈ M mn ( F )
∣ ∣ A ∣ ∣ F = ∑ j n ∑ i m ∣ a i j ∣ 2 . \vert \vert A \vert \vert_{F} = \sqrt{ \sum_{j}^{n} \sum_{i}^{m} \vert a_{ij} \vert ^{2} }. ∣∣ A ∣ ∣ F = j ∑ n i ∑ m ∣ a ij ∣ 2 . It is also given by
∣ ∣ A ∣ ∣ F = tr ( A † A ) . \vert \vert A \vert \vert_{F} = \sqrt{\textrm{tr}(A^{\dagger}A)}. ∣∣ A ∣ ∣ F = tr ( A † A ) . Let A ∈ M A~\in~M A ∈ M A ∈ M m n ( F ) A~\in~\mathbb{M}_{mn}(\mathbb{F}) A ∈ M mn ( F ) ψ ∈ V \psi~\in~V ψ ∈ V ψ ∈ M m n ( F ) \psi~\in~\mathbb{M}_{mn}(\mathbb{F}) ψ ∈ M mn ( F ) ( ψ 1 , ψ 2 , … , ψ n ) (\psi_1, \psi_2, \ldots, \psi_n) ( ψ 1 , ψ 2 , … , ψ n )
The l 2 l_2 l 2
∣ ∣ ψ ∣ ∣ 2 = ( ∣ ψ 1 ∣ 2 + ∣ ψ 2 ∣ 2 + … + ∣ ψ n ∣ 2 ) 1 / 2 \begin{align*}
\vert \vert \psi \vert \vert_{2} &= \big( \vert \psi_1 \vert^2 + \vert \psi_2 \vert^2 + \ldots + \vert \psi_n \vert^2)^{1/2}
\end{align*} ∣∣ ψ ∣ ∣ 2 = ( ∣ ψ 1 ∣ 2 + ∣ ψ 2 ∣ 2 + … + ∣ ψ n ∣ 2 ) 1/2 The l 2 l_2 l 2 operator norm l 2 l_2 l 2
∣ ∣ A ∣ ∣ 2 = sup ψ ∣ ∣ A ψ ∣ ∣ 2 ∣ ∣ ψ ∣ ∣ 2 \vert \vert A \vert \vert_2 = \sup_{\psi} \frac{ \vert \vert A \psi \vert \vert_2 }{ \vert \vert \psi \vert \vert_{2}} ∣∣ A ∣ ∣ 2 = ψ sup ∣∣ ψ ∣ ∣ 2 ∣∣ A ψ ∣ ∣ 2 It is also given by
∣ ∣ A ∣ ∣ 2 = μ max ( A † A ) \vert \vert A \vert \vert_2 = \sqrt{\mu_{\textrm{max}}(A^{\dagger}A)} ∣∣ A ∣ ∣ 2 = μ max ( A † A ) Let A ∈ M A~\in~M A ∈ M A ∈ M m n ( F ) A~\in~\mathbb{M}_{mn}(\mathbb{F}) A ∈ M mn ( F ) k k k
∣ ∣ A ∣ ∣ ∗ k = ∑ i k ∣ μ i ↓ ( A ) ∣ , \vert \vert A \vert \vert^{k} _{*} = \sum_{i}^{k} \vert \mu^{\downarrow}_{i}(A) \vert, ∣∣ A ∣ ∣ ∗ k = i ∑ k ∣ μ i ↓ ( A ) ∣ , where μ i ↓ ( A ) \mu^{\downarrow}_i (A) μ i ↓ ( A ) i i i A A A μ i ↓ ( A ) ≥ μ i + 1 ↓ ( A ) ∀ i \mu^{\downarrow}_i(A) \geq \mu^{\downarrow}_{i+1}(A)~\forall~i μ i ↓ ( A ) ≥ μ i + 1 ↓ ( A ) ∀ i k k k
Ky Fan Dominance : Let B ∈ M B~\in~M B ∈ M B ∈ M m n ( F ) B~\in~\mathbb{M}_{mn}(\mathbb{F}) B ∈ M mn ( F )
∣ ∣ A ∣ ∣ ∗ k ≤ ∣ ∣ B ∣ ∣ ∗ k , \vert \vert A \vert \vert^{k} _{*} \leq \vert \vert B \vert \vert^{k} _{*}, ∣∣ A ∣ ∣ ∗ k ≤ ∣∣ B ∣ ∣ ∗ k , for all k k k
∣ ∣ A ∣ ∣ ≤ ∣ ∣ B ∣ ∣ . \vert \vert A \vert \vert \leq \vert \vert B \vert \vert. ∣∣ A ∣∣ ≤ ∣∣ B ∣∣. Let A ∈ M A~\in~M A ∈ M A ∈ M m n ( F ) A~\in~\mathbb{M}_{mn}(\mathbb{F}) A ∈ M mn ( F )
∣ ∣ A ∣ ∣ p = ( ∑ i k [ μ i ( A ) ] p ) 1 p , \vert \vert A \vert \vert_{p} = \bigg( \sum_{i}^{k} \big[\mu_{i}(A)\big]^{p} \bigg)^{\frac{1}{p}}, ∣∣ A ∣ ∣ p = ( i ∑ k [ μ i ( A ) ] p ) p 1 , where μ i ( A ) \mu_i(A) μ i ( A ) A A A
Equally, Schatten norms can be written as
∣ ∣ A ∣ ∣ p = tr [ ∣ A ∣ p ] 1 p , \vert \vert A \vert \vert_{p} = \textrm{tr} \big[ \vert A \vert ^{p} \big]^{\frac{1}{p}}, ∣∣ A ∣ ∣ p = tr [ ∣ A ∣ p ] p 1 , where ∣ A ∣ = A A † \vert A \vert = \sqrt{AA^{\dagger}} ∣ A ∣ = A A †
If p = 1 p=1 p = 1 trace norm p = 2 p=2 p = 2 Frobenius Norm p = ∞ p=\infty p = ∞ l 2 l_{2} l 2
The following norms do not necessarily depend on the singular values and hence the above conditions might not hold.
This is an induced norm.
Let A ∈ M A~\in~M A ∈ M A ∈ F m × n A~\in~\mathbb{F}^{m \times n} A ∈ F m × n A A A A : V → W A: V \rightarrow W A : V → W V V V F m \mathbb{F}^{m} F m W W W F n \mathbb{F}^{n} F n
∣ ∣ A ∣ ∣ o p = inf { c ≥ 0 : ∣ ∣ A ψ ∣ ∣ W ≤ c ∣ ∣ ψ ∣ ∣ V ∀ ψ ∈ V } , \vert \vert A \vert \vert_{op} = \inf \{c \geq 0: \vert \vert A \psi \vert \vert_{W} \leq c \vert \vert \psi \vert \vert_{V}~\forall~\psi~\in~V \}, ∣∣ A ∣ ∣ o p = inf { c ≥ 0 : ∣∣ A ψ ∣ ∣ W ≤ c ∣∣ ψ ∣ ∣ V ∀ ψ ∈ V } , where ∣ ∣ ⋅ ∣ ∣ W \vert \vert \cdot \vert \vert_{W} ∣∣ ⋅ ∣ ∣ W ∣ ∣ ⋅ ∣ ∣ V \vert \vert \cdot \vert \vert_{V} ∣∣ ⋅ ∣ ∣ V V V V W W W
The operator norm can be colloquially thought of as the maximum amount a map A A A V V V
Let A ∈ M A~\in~M A ∈ M A ∈ F m × n A~\in~\mathbb{F}^{m \times n} A ∈ F m × n L p , q L_{p,q} L p , q p , q ≥ 1 p,q \geq 1 p , q ≥ 1
∣ ∣ A ∣ ∣ p , q = ( ∑ j n ( ∑ i m ∣ a i j ∣ p ) q p ) 1 q , \vert \vert A \vert \vert_{p,q} = \biggl( \sum^{n}_{j} \bigg( \sum_{i}^{m} \vert a_{ij} \vert^{p} \bigg)^{\frac{q}{p}} \biggl)^{\frac{1}{q}}, ∣∣ A ∣ ∣ p , q = ( j ∑ n ( i ∑ m ∣ a ij ∣ p ) p q ) q 1 , where a i j a_{ij} a ij A A A
The L 2 , 1 L_{2,1} L 2 , 1 l 2 l_{2} l 2 A A A
∣ ∣ A ∣ ∣ 2 , 1 = ∑ j n ∣ ∣ a j ∣ ∣ 2 = ∑ j n ( ∑ i m ∣ a i j ∣ 2 ) 1 2 \vert \vert A \vert \vert_{2,1} = \sum^{n}_{j} \vert \vert a_j \vert \vert_{2} = \sum^{n}_{j} \bigg( \sum_{i}^{m} \vert a_{ij} \vert^{2} \bigg)^{\frac{1}{2}} ∣∣ A ∣ ∣ 2 , 1 = j ∑ n ∣∣ a j ∣ ∣ 2 = j ∑ n ( i ∑ m ∣ a ij ∣ 2 ) 2 1 where a j a_{j} a j A A A
Cross Norms ¶ Consider a tensor product space V = X ⊗ Y V = \mathcal{X} \otimes \mathcal{Y} V = X ⊗ Y
A norm, ∣ ∣ ⋅ ∣ ∣ \vert \vert \cdot \vert \vert ∣∣ ⋅ ∣∣
∣ ∣ V ∣ ∣ = ∣ ∣ X ⊗ Y ∣ ∣ = ∣ ∣ X ∣ ∣ ∣ ∣ Y ∣ ∣ . \vert \vert V \vert \vert = \vert \vert \mathcal{X} \otimes \mathcal{Y} \vert \vert = \vert \vert \mathcal{X} \vert \vert ~ \vert \vert \mathcal{Y} \vert \vert. ∣∣ V ∣∣ = ∣∣ X ⊗ Y ∣∣ = ∣∣ X ∣∣ ∣∣ Y ∣∣. All norms that depend only on the singular values are cross norms.
Note : this is a somewhat hand-wavey proof.
Let the singular valued decomposition of a matrix A ∈ M n m ( C ) A~\in~\mathbb{M}_{nm}(\mathbb{C}) A ∈ M nm ( C )
A = U D A V † , A = U D_A V^{\dagger}, A = U D A V † , and the singular valued decomposition of a matrix B ∈ M n m ( C ) B~\in~\mathbb{M}_{nm}(\mathbb{C}) B ∈ M nm ( C )
B = W D B T † . B = W D_B T^{\dagger}. B = W D B T † . Hence, the singular values of A ⊗ B A \otimes B A ⊗ B A A A B B B
The singular valued decomposition of a matrix A ⊗ B A \otimes B A ⊗ B
A ⊗ B = ( U ⊗ W ) ( D A ⊗ D B ) ( V ⊗ T ) † . \begin{align*}
A \otimes B = (U \otimes W)(D_A \otimes D_B)(V \otimes T)^\dagger.
\end{align*} A ⊗ B = ( U ⊗ W ) ( D A ⊗ D B ) ( V ⊗ T ) † . By observing this fact, it can be seen that any norms that depends only the singular values of a matrix can be factorised into a product of the norm applied to A A A B B B
Useful Norm Inequalities ¶
Let H \mathcal{H} H Hilbert space A , B ∈ H A, B \in \mathcal{H} A , B ∈ H
The following inequality:
∣ ∣ A B ∣ ∣ r ≤ ∣ ∣ A ∣ ∣ p ∣ ∣ B ∣ ∣ q , \vert \vert AB \vert \vert_r \leq \vert \vert A \vert \vert_p ~ \vert \vert B \vert \vert_q, ∣∣ A B ∣ ∣ r ≤ ∣∣ A ∣ ∣ p ∣∣ B ∣ ∣ q , where 1 p + 1 q = 1 r \frac{1}{p} + \frac{1}{q} = \frac{1}{r} p 1 + q 1 = r 1
Note, Hölder’s Inequality is more general than this, and applies to vector spaces other than Hilbert spaces and objects other than operators. See here for some details
Let A ∈ H 1 ⊗ H 2 A \in \mathcal{H}_1 \otimes \mathcal{H}_2 A ∈ H 1 ⊗ H 2 H \mathcal{H} H ∣ ∣ ⋅ ∣ ∣ p \vert \vert \cdot \vert \vert_p ∣∣ ⋅ ∣ ∣ p Schatten norm
∣ ∣ tr 1 [ A ] ∣ ∣ p ≤ [ d i m ( H 1 ) ] ( p − 1 ) / p ∣ ∣ A ∣ ∣ p . \vert \vert ~ \textrm{tr}_1 [ A ] ~ \vert \vert_p \leq \big[ {\rm dim}(\mathcal{H}_1)]^{(p-1)/p} \vert \vert A \vert \vert_p. ∣∣ tr 1 [ A ] ∣ ∣ p ≤ [ dim ( H 1 ) ] ( p − 1 ) / p ∣∣ A ∣ ∣ p . The result it more general than this and more details can be found here
Let x , y ∈ V \bm{x}, \bm{y}~\in~V x , y ∈ V V V V inner product space F \mathbb{F} F ( ⋅ , ⋅ ) → F 1 (\cdot, \cdot) \rightarrow \mathbb{F}^{1} ( ⋅ , ⋅ ) → F 1
∣ ( x , y ) ∣ 2 ≤ ∣ ∣ x ∣ ∣ ∣ ∣ y ∣ ∣ , \vert (\bm{x}, \bm{y}) \vert ^{2} \leq \vert \vert \bm{x} \vert \vert ~ \vert \vert \bm{y} \vert \vert, ∣ ( x , y ) ∣ 2 ≤ ∣∣ x ∣∣ ∣∣ y ∣∣ , where ∣ ⋅ ∣ \vert \cdot \vert ∣ ⋅ ∣
∣ ∣ x ∣ ∣ = ( x , x ) . \vert \vert \bm{x} \vert \vert = \sqrt{ (\bm{x}, \bm{x}) }. ∣∣ x ∣∣ = ( x , x ) . Dual Norms ¶ Consider a normed vector space V V V ∣ ∣ ⋅ ∣ ∣ \vert \vert \cdot \vert \vert ∣∣ ⋅ ∣∣
There also exists norms on the dual space V V V V ∗ V^* V ∗ V V V
∣ ∣ f ∣ ∣ ∗ ≔ sup X { ∣ f ( X ) ∣ : ∣ ∣ X ∣ ∣ ≤ 1 , X ∈ V } , \vert \vert f \vert \vert^* \coloneqq \sup_{X} \big\{ \vert f(X) \vert : \vert \vert X \vert \vert \leq 1, ~ ~ X \in V \big\}, ∣∣ f ∣ ∣ ∗ : = X sup { ∣ f ( X ) ∣ : ∣∣ X ∣∣ ≤ 1 , X ∈ V } , where f ∈ V ∗ f \in V^* f ∈ V ∗ X ∈ V X \in V X ∈ V X X X f f f
These norms can be written in terms of only the primal vector space V V V
Consider a finite dimension vector space V V V inner product ( ⋅ , ⋅ ) (\cdot, \cdot) ( ⋅ , ⋅ )
The Riesz Representation Theorem then states that for all linear functionals f ∈ V ∗ f \in V^* f ∈ V ∗ Y ∈ V Y \in V Y ∈ V
f ( X ) = ( X , Y ) , X ∈ V . f(X) = (X, Y), ~ ~ X \in V. f ( X ) = ( X , Y ) , X ∈ V . Using the Riesz Representation Theorem, one can identify each linear functional f ∈ V ∗ f \in V^* f ∈ V ∗ Y ∈ V Y \in V Y ∈ V
∣ ∣ Y ∣ ∣ ∗ ≔ sup X { ∣ ( X , Y ) ∣ : ∣ ∣ X ∣ ∣ ≤ 1 , X ∈ V } , \vert \vert Y \vert \vert^* \coloneqq \sup_{X} \big\{ \vert (X,Y) \vert : \vert \vert X \vert \vert \leq 1, ~ ~ X \in V \big\}, ∣∣ Y ∣ ∣ ∗ : = X sup { ∣ ( X , Y ) ∣ : ∣∣ X ∣∣ ≤ 1 , X ∈ V } , where f ( X ) = ( X , Y ) f(X) = (X,Y) f ( X ) = ( X , Y ) Y Y Y V V V V ∗ V^* V ∗
For the inner product ( X , Y ) = tr [ X † Y ] (X,Y) = \textrm{tr} \big[ X^\dagger Y \big] ( X , Y ) = tr [ X † Y ]
∣ ∣ Y ∣ ∣ ∗ ≔ sup X { ∣ tr [ X † Y ] ∣ : ∣ ∣ X ∣ ∣ ≤ 1 , X ∈ V } . \vert \vert Y \vert \vert^* \coloneqq \sup_{X} \big\{ \vert \textrm{tr} \big[ X^\dagger Y \big] \vert : \vert \vert X \vert \vert \leq 1, ~ ~ X \in V \big\}. ∣∣ Y ∣ ∣ ∗ : = X sup { ∣ tr [ X † Y ] ∣ : ∣∣ X ∣∣ ≤ 1 , X ∈ V } . Dual Norm Examples ¶
The dual of the trace norm (one norm), ∣ ∣ A ∣ ∣ 1 \vert \vert A \vert \vert_1 ∣∣ A ∣ ∣ 1
∣ ∣ A ∣ ∣ 1 ∗ = ∣ ∣ A ∣ ∣ ∞ , \vert \vert A \vert \vert_1^* = \vert \vert A \vert \vert_{\infty}, ∣∣ A ∣ ∣ 1 ∗ = ∣∣ A ∣ ∣ ∞ , such that the dual norm of the trace norm is the infinity norm.
The dual of the infinity norm (operator norm), ∣ ∣ A ∣ ∣ ∞ \vert \vert A \vert \vert_{\infty} ∣∣ A ∣ ∣ ∞
∣ ∣ A ∣ ∣ ∞ ∗ = ∣ ∣ A ∣ ∣ 1 \vert \vert A \vert \vert_{\infty}^* = \vert \vert A \vert \vert_{1} ∣∣ A ∣ ∣ ∞ ∗ = ∣∣ A ∣ ∣ 1 such that the dual norm of the infinity norm is the trace norm.
Note, some norms as self dual, meaning the dual of the norm returns the same norm.
The Dual of a Dual ¶ The dual of a dual norm returns the original norm.
This can be used to give alternative ways to phrase the original norm. For example, the dual of the trace norm is defined as
∣ ∣ Y ∣ ∣ t r ∗ ≔ sup X { ∣ ( X , Y ) ∣ : ∣ ∣ X ∣ ∣ t r ≤ 1 , X ∈ V } . \vert \vert Y \vert \vert_{\rm tr}^* \coloneqq \sup_{X} \big\{ \vert (X,Y) \vert : \vert \vert X \vert \vert_{\rm tr} \leq 1, ~ ~ X \in V \big\}. ∣∣ Y ∣ ∣ tr ∗ : = X sup { ∣ ( X , Y ) ∣ : ∣∣ X ∣ ∣ tr ≤ 1 , X ∈ V } . Now, the dual of ∣ ∣ Y ∣ ∣ t r ∗ \vert \vert Y \vert \vert_{\rm tr}^* ∣∣ Y ∣ ∣ tr ∗
[ ∣ ∣ Y ∣ ∣ t r ∗ ] ∗ ≔ sup X { ∣ ( X , Y ) ∣ : ∣ ∣ X ∣ ∣ t r ∗ ≤ 1 , X ∈ V } . \big[ \vert \vert Y \vert \vert_{\rm tr}^* \big]^* \coloneqq \sup_{X} \big\{ \vert (X,Y) \vert : \vert \vert X \vert \vert_{\rm tr}^* \leq 1, ~ ~ X \in V \big\}. [ ∣∣ Y ∣ ∣ tr ∗ ] ∗ : = X sup { ∣ ( X , Y ) ∣ : ∣∣ X ∣ ∣ tr ∗ ≤ 1 , X ∈ V } . But, the dual of the dual is the original norm, such that
[ ∣ ∣ Y ∣ ∣ t r ∗ ] ∗ = ∣ ∣ Y ∣ ∣ t r . \big[ \vert \vert Y \vert \vert_{\rm tr}^* \big]^* = \vert \vert Y \vert \vert_{\rm tr}. [ ∣∣ Y ∣ ∣ tr ∗ ] ∗ = ∣∣ Y ∣ ∣ tr . It was then shown in the examples above that the dual of the trace norm is the infinity norm. Hence, an equivalent form for the trace norm is
∣ ∣ Y ∣ ∣ t r ≔ sup X { ∣ ( X , Y ) ∣ : ∣ ∣ X ∣ ∣ ∞ ≤ 1 , X ∈ V } . \vert \vert Y \vert \vert_{\rm tr} \coloneqq \sup_{X} \big\{ \vert (X,Y) \vert : \vert \vert X \vert \vert_{\infty} \leq 1, ~ ~ X \in V \big\}. ∣∣ Y ∣ ∣ tr : = X sup { ∣ ( X , Y ) ∣ : ∣∣ X ∣ ∣ ∞ ≤ 1 , X ∈ V } .
Rastegin, A. E. (2012). Relations for certain symmetric norms and anti-norms before and after partial trace . 10.48550/ARXIV.1202.3853