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Page "Banach algebra" ¶ 37
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norm and normal
Some investigators have found a parallelism between remissions and return of the sympathetic reactivity of the hypothalamus to the normal level as indicated by the Mecholyl test and, conversely, between clinical impairment and increasing deviation of this test from the norm.
Such accusations follow the breaking of some social norm, such as the failure to return a borrowed item, and any person part of the normal social exchange could potentially fall under suspicion.
Several other roots have similar normative and geometric descendants, such as Latin norma, whence norm, normal, and normative itself, and also geometric concepts such as normal vectors ; and likewise Greek ortho and Latin ordo, meaning either " right " or " correct " ( as in orthodox, meaning " correct opinion ") or " straight " or " perpendicular " ( as in orthogonal, meaning " perpendicular angle "), and thence order, ordinary, etc.
In other cultures, however, receptive fellatio is the norm for early adolescence and seen as a requirement for developing normal manliness.
Starting in the 1990s, hardware acceleration for normal consumer desktop computers has become the norm.
The operator norm of a normal matrix N equals the spectral and numerical radii of N. ( This fact generalizes to normal operators.
The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm ( x ) by software languages, as they differ only by scaling and translation.
The operator norm of a normal operator equals its numerical radius and spectral radius.
nancy boy – narcissism – narratophilia – nasophilia – natal sex – native lovemap – natural birth control – natural childbirth – natural family planning – natural law – navel fetish – navicular fossa of male urethra – Making out – necrobestiality – necromania – necrophagia – necrophilia – necrosadism – necrozoophilia – needle play – Sheikh Nefzaoui – neg ( seduction ) – Negotiation ( BDSM ) – Neisseria gonorrhoeae – neocortex – neolocal residence – neopagan marriage – neopagan views of marriage – neopagan wedding – neopenis – neophalloplasty – neophilia – nepiophilia – nepiphtherosis – NGU – niddah – nikah – nipple – nipple clamps – nipple covers – nipple discharge – nipple piercing – nipple reconstruction – nipple tassels – nitrite inhalants – no prostitutes here – no-fault divorce – no-pan kissa – nocturnal emission – non-gonococcal urethritis – non-specific urethritis – non-western concepts of male sexuality – nonce ( slang ) – nonmonogamy – normnormal ( behavior ) – normal sexuality – normophilia – normophiliac – Norplant – nosophilia – NSU – Nubian wedding – nuclear family – nude – nude chat – nudism – nudity – nulligravida – nullipara – nulliparous – nunsploitation – nutrition and pregnancy – nymph – nymphae – nymphectomy – nympholepsy – nymphomania – nymphophilia – nymphotomy –
Freud, among others, argued that neither predominantly different-nor same-sex sexuality were the norm, instead that what is called " bisexuality " is the normal human condition thwarted by society.
So long as these enhancements remain within a perceived norm of human behavior, a negative reaction is unlikely, but once individuals supplant normal human variety, revulsion can be expected.
Conversely the normal curvature is the norm of the projection of on the normal bundle to the submanifold at the point considered.
For the problem of solving the linear equation where is invertible, the condition number, where is the operator norm subordinate to the normal Euclidian norm on.
In some places, separate changing rooms for men and women are the norm, while elsewhere, a single change room with cubicles is normal.
Before the use of electronic transfers of payments became the norm in the United Kingdom the fortnightly ' giro ' payment was the normal way of distributing benefit payments.
Research by John Fleming and Jim Asplund indicates that engaged customers generate 1. 7 times more revenue than normal customers, while having engaged employees and engaged customers returns a revenue gain of 3. 4 times the norm.
This structure was the norm for all trains, normal or express, until 1953.
nation state — nationalism — nature — neocolonialism — neoliberalism — neo-locality — new international division of labour — non-state actor — non-tariff barriers to trade — normnormal type — normlessness — nuclear family
The word normal is used in a more narrow sense in mathematics, where a normal distribution describes a population whose characteristics centers around the average or the norm.

norm and element
A Banach algebra is called " unital " if it has an identity element for the multiplication whose norm is 1, and " commutative " if its multiplication is commutative.
A functional defined on some appropriate space of functions with norm < ref group =" Note "> The dot in this norm expression is a placeholder for an element of V, e. g. .</ ref > is said to have a weak minimum at the function if there exists some such that, for all functions with < math >
These algebras all have an involution ( or conjugate ), with the product of an element and its conjugate ( or sometimes the square root of this ) called the norm.
For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element.
For example, in the quaternion algebra H, the element t + x i + y j + z k has reduced norm t < sup > 2 </ sup > + x < sup > 2 </ sup > + y < sup > 2 </ sup > + z < sup > 2 </ sup > and reduced trace 2t.
If L / K is a Galois extension, the norm N < sub > L / K </ sub > of an element α of L is the product of all the conjugates
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension.
The definition is thus also compatible with norm of an element:
The norm of an ideal can be used to bound the norm of some nonzero element by the inequality
generated by an element s and if a is an element of L of relative norm 1, then there exists b in L such that
In number theory, the Hasse norm theorem states that if L / K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm.
Here to be a global norm means to be an element k of K such that there is an element l of L with ; in other words k is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then k is a norm from L < sub > P </ sub >; here the " prime " p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean.

norm and x
Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.
for all x and y in X, and where is the norm on X.
* Take the Banach space R < sup > n </ sup > ( or C < sup > n </ sup >) with norm || x ||
where is a norm on the domain / codomain of ƒ ( x ).
For every, there exists a polynomial function p over C such that for all x in, we have, or equivalently, the supremum norm.
If X and Y are subsets of the real numbers, d < sub > 1 </ sub > and d < sub > 2 </ sub > can be the standard Euclidean norm, || · ||, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all x, y ∈ X, | x − y | < δ implies | f ( x ) − f ( y )| < ε.
If X is a normed space, then the dual space X * is itself a normed vector space by using the norm ǁφǁ = sup < sub > ǁxǁ ≤ 1 </ sub >| φ ( x )|.
A closely related notion with geometric overtones is a quadratic space, which is a pair ( V, q ), with V a vector space over a field K, and a quadratic form on V. An example is given by the three-dimensional Euclidean space and the square of the Euclidean norm expressing the distance between a point with coordinates ( x, y, z ) and the origin:
C (| s − s < nowiki >'</ nowiki >|) where | x | denotes the norm of the vector x ( for actual rotations this is the Euclidean or 2-norm ).
Note that π ( x ) is a bounded linear operator since it is the direct sum of a family of operators, each one having norm ≤ || x ||.
must be positive definite at x = 0 by assumption 5, the particle states have positive norm.
If x is orthogonal to y, then and the above equation for the norm of a sum becomes:
where F ( x, · ) is a Minkowski norm ( or at least an asymmetric norm ) on each tangent space T < sub > x </ sub > M.

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