In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.

Definition

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Let   be a normed vector space with norm   and let   denote its continuous dual space. The dual norm of a continuous linear functional   belonging to   is the non-negative real number defined[1] by any of the following equivalent formulas:   where   and   denote the supremum and infimum, respectively. The constant   map is the origin of the vector space   and it always has norm   If   then the only linear functional on   is the constant   map and moreover, the sets in the last two rows will both be empty and consequently, their supremums will equal   instead of the correct value of  

Importantly, a linear function   is not, in general, guaranteed to achieve its norm   on the closed unit ball   meaning that there might not exist any vector   of norm   such that   (if such a vector does exist and if   then   would necessarily have unit norm  ). R.C. James proved James's theorem in 1964, which states that a Banach space   is reflexive if and only if every bounded linear function   achieves its norm on the closed unit ball.[2] It follows, in particular, that every non-reflexive Banach space has some bounded linear functional that does not achieve its norm on the closed unit ball. However, the Bishop–Phelps theorem guarantees that the set of bounded linear functionals that achieve their norm on the unit sphere of a Banach space is a norm-dense subset of the continuous dual space.[3][4]

The map   defines a norm on   (See Theorems 1 and 2 below.) The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces. Since the ground field of   (  or  ) is complete,   is a Banach space. The topology on   induced by   turns out to be stronger than the weak-* topology on  

The double dual of a normed linear space

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The double dual (or second dual)   of   is the dual of the normed vector space  . There is a natural map  . Indeed, for each   in   define  

The map   is linear, injective, and distance preserving.[5] In particular, if   is complete (i.e. a Banach space), then   is an isometry onto a closed subspace of  .[6]

In general, the map   is not surjective. For example, if   is the Banach space   consisting of bounded functions on the real line with the supremum norm, then the map   is not surjective. (See   space). If   is surjective, then   is said to be a reflexive Banach space. If   then the space   is a reflexive Banach space.

Examples

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Dual norm for matrices

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The Frobenius norm defined by   is self-dual, i.e., its dual norm is  

The spectral norm, a special case of the induced norm when  , is defined by the maximum singular values of a matrix, that is,   has the nuclear norm as its dual norm, which is defined by   for any matrix   where   denote the singular values[citation needed].

If   the Schatten  -norm on matrices is dual to the Schatten  -norm.

Finite-dimensional spaces

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Let   be a norm on   The associated dual norm, denoted   is defined as  

(This can be shown to be a norm.) The dual norm can be interpreted as the operator norm of   interpreted as a   matrix, with the norm   on  , and the absolute value on  :  

From the definition of dual norm we have the inequality   which holds for all   and  [7] The dual of the dual norm is the original norm: we have   for all   (This need not hold in infinite-dimensional vector spaces.)

The dual of the Euclidean norm is the Euclidean norm, since  

(This follows from the Cauchy–Schwarz inequality; for nonzero   the value of   that maximises   over   is  )

The dual of the  -norm is the  -norm:   and the dual of the  -norm is the  -norm.

More generally, Hölder's inequality shows that the dual of the  -norm is the  -norm, where   satisfies   that is,  

As another example, consider the  - or spectral norm on  . The associated dual norm is   which turns out to be the sum of the singular values,   where   This norm is sometimes called the nuclear norm.[8]

Lp and ℓp spaces

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For   p-norm (also called  -norm) of vector   is  

If   satisfy   then the   and   norms are dual to each other and the same is true of the   and   norms, where   is some measure space. In particular the Euclidean norm is self-dual since   For  , the dual norm is   with   positive definite.

For   the  -norm is even induced by a canonical inner product   meaning that   for all vectors   This inner product can expressed in terms of the norm by using the polarization identity. On   this is the Euclidean inner product defined by   while for the space   associated with a measure space   which consists of all square-integrable functions, this inner product is   The norms of the continuous dual spaces of   and   satisfy the polarization identity, and so these dual norms can be used to define inner products. With this inner product, this dual space is also a Hilbert spaces.

Properties

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Given normed vector spaces   and   let  [9] be the collection of all bounded linear mappings (or operators) of   into   Then   can be given a canonical norm.

Theorem 1 — Let   and   be normed spaces. Assigning to each continuous linear operator   the scalar   defines a norm   on   that makes   into a normed space. Moreover, if   is a Banach space then so is  [10]

Proof

A subset of a normed space is bounded if and only if it lies in some multiple of the unit sphere; thus   for every   if   is a scalar, then   so that  

The triangle inequality in   shows that  

for every   satisfying   This fact together with the definition of   implies the triangle inequality:  

Since   is a non-empty set of non-negative real numbers,   is a non-negative real number. If   then   for some   which implies that   and consequently   This shows that   is a normed space.[11]

Assume now that   is complete and we will show that   is complete. Let   be a Cauchy sequence in   so by definition   as   This fact together with the relation  

implies that   is a Cauchy sequence in   for every   It follows that for every   the limit   exists in   and so we will denote this (necessarily unique) limit by   that is:  

It can be shown that   is linear. If  , then   for all sufficiently large integers n and m. It follows that   for sufficiently all large   Hence   so that   and   This shows that   in the norm topology of   This establishes the completeness of  [12]

When   is a scalar field (i.e.   or  ) so that   is the dual space   of  

Theorem 2 — Let   be a normed space and for every   let   where by definition   is a scalar. Then

  1.   is a norm that makes   a Banach space.[13]
  2. If   is the closed unit ball of   then for every     Consequently,   is a bounded linear functional on   with norm  
  3.   is weak*-compact.
Proof

Let  denote the closed unit ball of a normed space   When   is the scalar field then   so part (a) is a corollary of Theorem 1. Fix   There exists[14]   such that   but,   for every  . (b) follows from the above. Since the open unit ball   of   is dense in  , the definition of   shows that   if and only if   for every  . The proof for (c)[15] now follows directly.[16]

As usual, let   denote the canonical metric induced by the norm on   and denote the distance from a point   to the subset   by   If   is a bounded linear functional on a normed space   then for every vector  [17]   where   denotes the kernel of  

See also

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Notes

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  1. ^ Rudin 1991, p. 87
  2. ^ Diestel 1984, p. 6.
  3. ^ Bishop, Errett; Phelps, R. R. (1961). "A proof that every Banach space is subreflexive". Bulletin of the American Mathematical Society. 67: 97–98. doi:10.1090/s0002-9904-1961-10514-4. MR 0123174.
  4. ^ Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel Journal of Mathematics. 115: 25–28. doi:10.1007/bf02810578. MR 1749671. S2CID 53646715.
  5. ^ Rudin 1991, section 4.5, p. 95
  6. ^ Rudin 1991, p. 95
  7. ^ This inequality is tight, in the following sense: for any   there is a   for which the inequality holds with equality. (Similarly, for any   there is an   that gives equality.)
  8. ^ Boyd & Vandenberghe 2004, p. 637
  9. ^ Each   is a vector space, with the usual definitions of addition and scalar multiplication of functions; this only depends on the vector space structure of  , not  .
  10. ^ Rudin 1991, p. 92
  11. ^ Rudin 1991, p. 93
  12. ^ Rudin 1991, p. 93
  13. ^ Aliprantis & Border 2006, p. 230
  14. ^ Rudin 1991, Theorem 3.3 Corollary, p. 59
  15. ^ Rudin 1991, Theorem 3.15 The Banach–Alaoglu theorem algorithm, p. 68
  16. ^ Rudin 1991, p. 94
  17. ^ Hashimoto, Nakamura & Oharu 1986, p. 281.

References

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