In general, we write ∥ ⋅, …, ⋅ ∥ X, and for a standard case, we write ∥ ⋅, …, ⋅ ∥ S. For a derived norm, we use ∥ ⋅, …, ⋅ ∥ ∞.
Then a sequence in Y is convergent in the norm ∥ ⋅ ∥ ( s ) 0 defined by (2.1) if and only if it is convergent in the derived norm ∥ ⋅ ∥ ( s ) 1 and, by induction, in the derived norm ∥ ⋅ ∥ ( s ) r defined by (2.3) for all r = 1, …, n − 1.
By induction, Y is complete with respect to the norm ∥ ⋅ ∥ ( s ) 0 if and only if it is complete with respect to the derived norm ∥ ⋅ ∥ ( s ) n − 1 defined by (2.4).
If X is equipped with the standard n-norm and the derived norm is with respect to an orthonormal set, then the converse of the above theorem is also true.
Then ( x i ) is convergent in Y in the norm ∥ ⋅ ∥ ( s ) 0 defined by (2.1) if and only if ( x i ) is convergent in Y in the derived norm ∥ ⋅ ∥ ( s ) r defined by (2.3) for r = 1.
Theorem 2.7 If ( x i ) converges to an x in Y in the norm ∥ ⋅ ∥ ( s ) 0 defined by (2.1), then ( x i ) also converges to x in the derived norm ∥ ⋅ ∥ ( s ) r defined by (2.3) for r = 1.
Then Y is complete with respect to the norm ∥ ⋅ ∥ ( s ) 0 defined by (2.1) if and only if it is complete with respect to the derived norm ∥ ⋅ ∥ ( s ) 1 defined by (2.3).
In particular, a sequence in Y is convergent in the norm ∥ ⋅ ∥ ( s ) 0 if and only if it is convergent in the derived norm ∥ ⋅ ∥ ( s ) n − 1, defined by ∥ x ∥ ( s ) n − 1 = inf { ρ > 0 : sup k M k ( ∥ Δ ( s ) m x k ρ ∥ ∞ ) ≤ 1 }.
This paper describes a procedure for deriving norm-bounded output-multiplicative uncertainty descriptions for a multi-input multi-output system by matching the output of an uncertainty model to the outputs of a set of known models.
In this section, we consider the two accounts of epistemic disutility for credences given in the previous section and we combine them with decision-theoretic norms to derive epistemic norms.
