A firmly non-expansive mapping is always non-expansive, via the Cauchy – Schwarz inequality.

A common German folk-tale is of the German priest / monk named Berthold Schwarz who independently invented gunpowder, thus earning it the German name Schwarzpulver or in English Schwarz's powder.

In mathematics, the Cauchy – Schwarz inequality ( also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy – Bunyakovsky – Schwarz inequality, or Cauchy – Bunyakovsky inequality ), is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas.

The Cauchy – Schwarz inequality states that for all vectors x and y of an inner product space it is true that

In Euclidean space R < sup > n </ sup > with the standard inner product, the Cauchy – Schwarz inequality is

The triangle inequality for the inner product is often shown as a consequence of the Cauchy – Schwarz inequality, as follows: given vectors x and y:

The Cauchy – Schwarz inequality proves that this definition is sensible, by showing that the right hand side lies in the interval, and justifies the notion that ( real ) Hilbert spaces are simply generalizations of the Euclidean space.

The Cauchy – Schwarz is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself.

The Cauchy – Schwarz inequality is usually used to show Bessel's inequality.

One can see from the proof that the Cauchy – Schwarz inequality is a consequence of the positivity and anti-symmetry inner-product axioms.

This is precisely the Cauchy – Schwarz inequality.

When Φ is 2-positive, a stronger assumption than merely positive, one has something that looks very similar to the original Cauchy – Schwarz inequality:

The general formulation of the Heisenberg uncertainty principle is derived using the Cauchy – Schwarz inequality in the Hilbert space of quantum observables.

An English translation of part of Slavata's report of the incident is printed in Henry Frederick Schwarz, The Imperial Privy Council in the Seventeenth Century ( Cambridge, Mass.

German anatomist Ernst Schwarz is credited with having discovered the bonobo in 1928, based on his analysis of a skull in the Tervuren museum in Belgium that previously had been thought to have belonged to a juvenile chimpanzee.

Schwarz ( an adjective ) is German for " black ".

In the first episode of the animated series, the Dark Side is called " The Schwarz side of the Schwartz ".

Yet another explanation is that it is a reference to FAO Schwarz, which sells lots of movie merchandise.

It is then enough to minimize the functional E, owing to the Cauchy – Schwarz inequality

Teenburger also provided record sleeve designs for the bands Brinsley Schwarz and Red Dirt, as well as Vertigo artists such as Cressida, Gracious!

When n = 3 the Cauchy – Schwarz inequality can also be deduced from Lagrange's identity, which takes the form

The RLPO's catalogue also includes a symphony cycle and other works by Beethoven with Sir Charles Mackerras, Britten, Mahler symphonies with Schwarz, Pešek and Mackerras, as well as many works of Rachmaninov, Smetana, Richard Strauss with those conductors.

Together with David Schkade, Kahneman developed the notion of the focusing illusion ( Kahneman & Schkade, 1998 ; Kahneman, Krueger, Schkade, Schwarz & Stone, 2006 ) to explain in part the mistakes people make when estimating the effects of different scenarios on their future happiness ( also known as affective forecasting, which has been studied extensively by Daniel Gilbert ).

In addition to the albums, Brinsley Schwarz also issued a series of singles under their own name, and various pseudonyms, such as " The Hitters ", " The Knees ", " Limelight " and " The Brinsleys " but these all failed.

The original version of Elvis Costello's hit "( What's So Funny ' Bout ) Peace, Love, and Understanding ", written by Lowe, was released by the band in 1974 on their album The New Favourites of Brinsley Schwarz, and also featured on the multimillion selling soundtrack of the movie The Bodyguard, where it was performed by Curtis Stigers.

Formed in February 1972, with ex Brinsley Schwarz roadie Martin Belmont on guitar, former Help Yourself collaborator Sean Tyla, also on guitar, ex Help Yourself bassist Ken Whaley, and Magic Michael ( Michael Cousins ) on percussion.

One of the former Tally Ho jazz players, Barry Richardson, was so impressed he formed Bees Make Honey, who were also given a Tally Ho residency, alongside a mix of new and existing bands, such as Max Merritt and the Meteors and Brinsley Schwarz.

FAO Schwarz also features limited-quantity luxury items including a $ 1, 500 Jeweled Classic Etch-A-Sketch and a Barbie Foosball Table by Bonzini priced at $ 25, 000.

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.

This may improve the sound, and also avoids the peril of warping — as harpsichord makers Kerstin Schwarz and Tony Chinnery point out,, a severely warped soundboard threatens a structural catastrophe, namely contact between strings and soundboard.

After learning the trade of producer, culminating in a couple of singles in the style of Phil Spector, " Baby I Love You " and " Born to Be with You ", he became linked with the pub rock movement of the early 1970s, producing Brinsley Schwarz, Ducks Deluxe, and also The Flamin ' Groovies, using a stripped down, grittier sound.

* Schwarz, an album by Conrad Schnitzler ; also called Eruption

In addition to the composer's own recording, those by Erich Kunzel and Gerard Schwarz are also popular.

He also returned to the study of functions in the 1940s, continuing his work on the extension problems formulated a decade earlier, and answering a question of Schwarz in a 1948 paper On Ideals of Differentiable Functions.

Toyota drivers Kankkunen, Auriol and Armin Schwarz were also stripped of all points in the championships.

He also continues to run his indie record label Midi: Nette, and has produced for artists such as Schwarz Stein and Kanon Wakeshima.

According to Johannes Valentin Schwarz, the events also encouraged the growth of the modern Jewish press.

Minnesota Nice is also the title of a 2003, 28-minute documentary by Jeffrey Schwarz about the Coen brothers ' movie Fargo, a movie that displays some of the stereotypical behavior.

K. Robert Schwarz has noted the influence of the musical techniques of Steve Reich on Harmonium, and also has commented on the less schematic and more " intuitive " manner of Adams ' composition in the work.

* The Fifth Enemy Offensive, also known as the Battle of the Sutjeska or Fall Schwarz ( Case Black ).

* 1989 – Heiko Schwarz, German footballer

* 1986 – Udo Schwarz, German rugby player

* 1812 – Karl Schwarz, German theologian ( d. 1885 )

The 16th century German banker Jakob Fugger and his principal accountant, M. Schwarz, registering an entry to a ledger.

Karl Schwarz ( 19 November 1812-25 March 1885 ) was a German Protestant theologian.

Schwarz took an important part in the founding and directing of the German Protestantenverein and became an eminent exponent of liberal theology.

FAO Schwarz was founded in 1862 under the name “ Toy Bazaar ” by German immigrant Frederick August Otto Schwarz, in Baltimore, where he and his brothers retailed toys from a fancy-goods store.

The German blazon reads: Schräglinks geteilt: vorne in Silber ein durchgehendes rotes Kreuz, belegt mit einem silbernem Hifthorn mit goldenem Riemen ; hinten in Schwarz ein wachsender, rot bewehrter, gezungter und gekrönter goldener Löwe.

Schwarz is a common surname, derived from the German,, meaning the color black.

* Annette Schwarz, German pornographic actress

* Christian Friedrich Schwarz, German Protestant missionary

* Ernst Schwarz ( 1889 – 1962 ), German zoologist who named the bonobo

* Frederick August Otto Schwarz ( 1836 – 1911 ), German immigrant founder of F. A. O.

* Gotthold Schwarz ( born 1952 ), German bass-baritone and conductor

* Hermann Schwarz ( philosopher ), German philosopher

* Hermann Amandus Schwarz ( 1843 – 1921 ), German mathematician

* Josef Schwarz ( born 1941 ), Czech-born German long jumper

* Karl Schwarz ( 1812 – 1885 ), German Protestant theologian