Norms and Dual Norms as Supremums and Infimums
Let be a finite-dimensional Hilbert space over or (the fields of real and complex numbers, respectively). If we let be a norm on (not necessarily the norm induced by the inner product), then the dual norm of is defined by
The double-dual of a norm is equal to itself (i.e., ) and the norm induced by the inner product is the unique norm that is its own dual. Similarly, if is the vector p-norm, then , where satisfies .
In this post, we will demonstrate that has an equivalent characterization as an infimum, and we use this characterization to provide a simple derivation of several known (but perhaps not well-known) formulas for norms such as the operator norm of matrices.
For certain norms (such as the “separability norms” presented at the end of this post), this ability to write a norm as both an infimum and a supremum is useful because computation of the norm may be difficult. However, having these two different characterizations of a norm allows us to bound it both from above and from below.
The Dual Norm as an Infimum
Theorem 1. Let be a bounded set satisfying and define a norm by
Then is given by
where the infimum is taken over all such decompositions of .
Before proving the result, we make two observations. Firstly, the quantity described by Theorem 1 really is a norm: boundedness of ensures that the supremum is finite, and ensures that . Secondly, every norm on can be written in this way: we can always choose to be the unit ball of the dual norm . However, there are times when other choices of are more useful or enlightening (as we will see in the examples).
Proof of Theorem 1. Begin by noting that if and then . It follows that whenever . In fact, we now show that is the largest norm on with this property. To this end, let be another norm satisfying whenever . Then
Thus , so by taking duals we see that , as desired.
For the remainder of the proof, we denote the infimum in the statement of the theorem by . Our goal now is to show that: (1) is a norm, (2) satisfies whenever , and (3) is the largest norm satisfying property (2). The fact that will then follow from the first paragraph of this proof.
To see (1) (i.e., to prove that is a norm), we only prove the triangle inequality, since positive homogeneity and the fact that if and only if are both straightforward (try them yourself!). Fix and let , be decompositions of with for all i, satisfying and . Then
Since was arbitrary, the triangle inequality follows, so is a norm.
To see (2) (i.e., to prove that whenever ), we simply write in its trivial decomposition , which gives the single coefficient , so .
To see (3) (i.e., to prove that is the largest norm on satisfying condition (2)), begin by letting be any norm on with the property that for all . Then using the triangle inequality for shows that if is any decomposition of with for all i, then
Taking the infimum over all such decompositions of shows that , which completes the proof.
The remainder of this post is devoted to investigating what Theorem 1 says about certain specific norms.
Injective and Projective Cross Norms
If we let , where and are themselves finite-dimensional Hilbert spaces, then one often considers the injective and projective cross norms on , defined respectively as follows:
where here refers to the norm induced by the inner product on or . The fact that and are duals of each other is simply Theorem 1 in the case when S is the set of product vectors:
In fact, the typical proof that the injective and projective cross norms are duals of each other is very similar to the proof of Theorem 1 provided above (see [1, Chapter 1]).
Maximum and Taxicab Norms
Use to denote the dimension of and let be an orthonormal basis of . If we let then the norm in the statement of Theorem 1 is the maximum norm (i.e., the p = ∞ norm):
where is the i-th coordinate of in the basis . The theorem then says that the dual of the maximum norm is
which is the taxicab norm (i.e., the p = 1 norm), as we expect.
Operator and Trace Norm of Matrices
If we let , the space of complex matrices with the Hilbert–Schmidt inner product
where is the Euclidean norm on . If we let be the set of unitary matrices in , then Theorem 1 provides the following alternate characterization of the operator norm:
Corollary 1. Let . Then
As an application of Corollary 1, we are able to provide the following characterization of unitarily-invariant norms (i.e., norms with the property that for all unitary matrices ):
Corollary 2. Let be a norm on . Then is unitarily-invariant if and only if
for all .
Proof of Corollary 2. The “if” direction is straightforward: if we let and be unitary, then
where we used the fact that . It follows that , so is unitarily-invariant.
To see the “only if” direction, write and with each and unitary. Then
By taking the infimum over all decompositions of and of the given form and using Corollary 1, the result follows.
An alternate proof of Corollary 2, making use of some results on singular values, can be found in [2, Proposition IV.2.4].
As our final (and least well-known) example, let , again with the usual Hilbert–Schmidt inner product. If we let
where is the Euclidean norm on or , then Theorem 1 tells us that the following two norms are dual to each other:
There’s actually a little bit of work to be done to show that has the given form, but it’s only a couple lines – consider it an exercise for the interested reader.
Both of these norms come up frequently when dealing with quantum entanglement. The norm was the subject of , where it was shown that a quantum state is entangled if and only if (I use the above duality relationship to provide an alternate proof of this fact in [4, Theorem 6.1.5]). On the other hand, the norm characterizes positive linear maps of matrices and was the subject of [5, 6].
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