* Take the Banach space R < sup > n </ sup > ( or C < sup > n </ sup >) with norm || x ||

The spectrum of any element x is a closed subset of the closed disc in C with radius || x || and center 0, and thus is compact.

* || x *||

* || x x *|| =

|| x ||< sup > 2 </ sup > for all x in A.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

The sphere is the inverse image of a one-point set under the continuous function || x ||.

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.

*|| Tx || = || T * x || for all x ( use ).

and after multiplication by || v ||< sup > 2 </ sup > the Cauchy – Schwarz inequality.

Here, || ||< sub > 2 </ sup > is the matrix 2-norm, c < sub > n </ sub > is a small constant depending on n, and ε denotes the unit round-off.

Hölder's inequality holds even if || fg ||< sub > 1 </ sub > is infinite, the right-hand side also being infinite in that case.

* If 1 ≤ p, q < ∞, then || f ||< sub > p </ sup > and || g ||< sub > q </ sup > stand for the ( possibly infinite ) expressions

* If p = ∞, then || f ||< sub >∞</ sup > stands for the essential supremum of | f |, similarly for || g ||< sub >∞</ sup >.

* The notation || f ||< sub > p </ sup > with 1 ≤ p ≤ ∞ is a slight abuse, because in general it is only a norm of f if || f ||< sub > p </ sup > is finite and f is considered as equivalence class of μ-almost everywhere equal functions.

As above, let f and g denote measurable real-or complex-valued functions defined on S. If || fg ||< sub > 1 </ sup > is finite, then the products of f with g and its complex conjugate function, respectively, are μ-integrable, the estimates

This is also called Cauchy – Schwarz inequality, but requires for its statement that || f ||< sub > 2 </ sup > and || g ||< sub > 2 </ sup > are finite to make sure that the inner product of f and g is well defined.

The magnitude || P ||< sup > 2 </ sup > is Lorentz invariant, meaning its value is not changed by Lorentz transformations / boosting into different frames of reference.

label ( M ) + log < sub > p </ sub > || det g ||< sub > p </ sub > modulo n.

Theorem For ƒ in L < sup > 1 </ sup >( R < sup > d </ sup >), the above series converges pointwise almost everywhere, and thus defines a periodic function Pƒ on Λ. Pƒ lies in L < sup > 1 </ sup >( Λ ) with || Pƒ ||< sub > 1 </ sub > ≤ || ƒ ||< sub > 1 </ sub >.

: Pt ( s ) | H < sub > 2 </ sub >( 1 atm ) | H < sup >+</ sup >( 1 M ) || Cu < sup > 2 +</ sup > ( 1 M ) | Cu ( s )

: Reference electrode | concentrated solution of KCl || test solution | H < sub > 2 </ sub > | Pt

| Queen || Elizabeth II || || 6 February 1952 </ tr >

| Queen's Representative || Sir Frederick Goodwin KBE || || 9 February 2001 </ tr >

| Prime Minister || Henry Puna || CIP || 30 November 2010 </ tr >

: Zn ( s ) | Zn < sup > 2 +</ sup > ( 1M ) || Cu < sup > 2 +</ sup > ( 1M ) | Cu ( s )

: Cu ( s ) | Cu < sup > 2 +</ sup > ( 0. 05 M ) || Cu < sup > 2 +</ sup > ( 2. 0 M ) | Cu ( s )

|| 2 ||” ( p 1259 ).

| John Vliet Lindsay || 1944 || 103rd Mayor of New York City 1966-1973 < p > Congressman from New York City 1959-1965.

( 2003 ) comments that the technology has gone through three phases: or < p > Off-Line Scan Uncorrectable Sector Count || alt = Lower || The total count of uncorrectable errors when reading / writing a sector.

| 201 || 0xC9 || Soft Read Error Rate or < p > TA Counter Detected || alt = Lower || Count of off-track errors.

| President || Manny Mori || || May 11, 2007 </ tr >

| Vice-President || Alik L. Alik || || May 11, 2007 </ tr >

< u > Hige </ u > sceal þe < u > heard </ u > ra, || < u > heor </ u > te þe < u > cēn </ u > re,

| rowspan =" 2 "| || rowspan =" 2 "| " Freak on a Leash " || Best Hard Rock Performance ||

| rowspan =" 9 "| || rowspan =" 9 "| " Freak on a Leash " || Best Rock Video ||

| 1999 || " Freak on a Leash " || Best International Video ||

| Code 16K || Based on 1D Code 128.

| d-touch || readable when printed on deformable gloves and stretched and distorted

| Nintendo e-Reader # Dot code || Developed by Olympus Corporation to store songs, images, and mini-games for Game Boy Advance on Pokémon trading cards.

The optimization problem presented in the preceding section is difficult to solve because it depends on || w ||, the norm of w, which involves a square root.

| style =" text-align: center ;"| 1997 || " Trippin ' on a Hole in a Paper Heart " || Grammy Award for Best Hard Rock Performance ||

* From the unit length restriction on u, || u || =

* From the unit length restriction on v, || v || = 1.

The bound || T || on the spectrum can be refined somewhat.

While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers and is given by ; then for a complex number its absolute value || coincides with its Euclidean norm, and its argument with the angle turning from 1 to.