As a consequence of the Hahn – Banach theorem, J is norm-preserving ( i. e., || J ( x )|| = || x || ) and hence injective.

He is the enigmatic dictator of Oceania, a totalitarian state taken to its utmost logical consequence – where the ruling Party wields total power for its own sake over the inhabitants.

Deane and J. R. Thomson write this valid conclusion, “ The Book of Obadiah is occupied with one subject – the punishment of Edom for its cruel and unbrotherly love conduct towards Judah ...” One can link this idea of punishment to one of the major prophets “ Ezekiel ” who “... interprets the exile to Babylon and the destruction of Jerusalem as deserved punishments for the sins of those who themselves committed them .” Verses 3-7 in Obadiah explain to the reader the reason for the punishment theme, “ Confidence in one ’ s power, intelligence, allies, or the topographical features of one ’ s territory is often mentioned as an attribute of those who foolishly confront the Lord and are consequently punished .” Although destruction is vital to understanding Obadiah, it is of note to understand the destruction being a consequence of action.

As a consequence of the Nyquist – Shannon sampling theorem, any spatial waveform that can be displayed must consist of at least two pixels, which is proportional to image resolution.

Block structure was introduced into computer programming languages by the Algol project ( 1957 – 1960 ), which, as a consequence, also featured a context-free grammar to describe the resulting Algol syntax.

His grandfather, Thomas Francis, founder of the Carignano line of the House of Savoy, was the son of Catherine Michelle – a daughter of Philip II of Spain – and the great-grandson of the Emperor Charles V. But of more immediate consequence to Leopold I was the fact that Eugene was the second cousin of Victor Amadeus, the Duke of Savoy, a connection that the Emperor hoped might prove useful in any future confrontation with France.

Even the " exact " FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors are typically quite small ; most FFT algorithms, e. g. Cooley – Tukey, have excellent numerical properties as a consequence of the pairwise summation structure of the algorithms.

As a consequence, channel length should not exceed 10 – 15 metres.

The H – H bond is one of the strongest bonds in chemistry, with a bond dissociation enthalpy of 435. 88 kJ / mol at 298 K. As a consequence of this strong bond, H < sub > 2 </ sub > dissociates to only a minor extent until higher temperatures.

* 1878 – League of Prizren is established, to oppose the decisions of the Congress of Berlin and the Treaty of San Stephano, as a consequence of which the Albanian lands in Balkans were being partitioned and given to the neighbor states of Serbia, Montenegro, Bulgaria and Greece.

* 1621 – Execution of 27 Czech noblemen on the Old Town Square in Prague as a consequence of the Battle of White Mountain.

As a direct consequence of the 1932 – 33 tour, the MCC introduced a new rule to the laws of cricket in 1935.

These included most famously Troio Savelli, scion of a powerful ancient Roman family, and the youthful and noble Beatrice Cenci, who had murdered her father – probably as a consequence of his repeated abuses.

Another consequence of the Molotov-Ribbentrop Pact was the Soviet occupation of Bessarabia and Northern Bukovina, carried out by the Southern Front in June – July 1940.

* create value – resources expended to mitigate risk should be less than the consequence of inaction, or ( as in value engineering ), the gain should exceed the pain

A consequence of the Jacobson density theorem is Wedderburn's theorem ; namely that any right artinian simple ring is isomorphic to a full matrix ring of n by n matrices over a division ring for some n. This can also be established as a corollary of the Artin – Wedderburn theorem.

St Augustine of Hippo ( 354 AD – 430 ) in his Augustinian theodicy focuses on the Genesis story that essentially dictates that God created the world and that it was good ; evil is merely a consequence of the fall of man ( The story of the Garden of Eden where Adam and Eve disobeyed God and caused inherent sin for man ).

* June 21 – Thirty Years ' War: Twenty-seven Czech lords are executed on the Old Town Square in Prague as a consequence of the Battle of White Mountain.

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:

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

# distinctness – in regards of evolutionarily relevant criteria, i. e. ecology, morphology, or biogeography ; note that DNA sequences are a consequence rather than a condition of diverging evolutionary lineages except in cases where they directly inhibit gene flow ( e. g. postzygotic barriers ).

Estonia had pursued a policy of neutrality, but it was of no consequence after the Soviet Union and Nazi Germany signed the Molotov – Ribbentrop Pact on August 23, 1939.

Also according to Herodotus ( III. 139 ), Darius, prior to seizing power and " of no consequence at the time ", had served as a spearman ( doryphoros ) in the Egyptian campaign ( 528 – 525 BCE ) of Cambyses II, then the Persian Great King.

* The Euclidean spaces R < sup >< var > n </ var ></ sup > ( and in particular the real line R ) are locally compact as a consequence of the Heine – Borel theorem.

In fact, the space of all Lipschitz functions on a compact metric space is dense in the Banach space of continuous functions, an elementary consequence of the Stone – Weierstrass theorem.

shows that, thus proving the global uniqueness ( the local uniqueness is a consequence of the uniqueness of the Banach fixed point ).

When the theory of distributions was still not widely known nor used, property above was formulated by saying that the convolution of the function ' with a given function belonging to a proper Hilbert or Banach space converges as ε → 0 to this last one: this is exactly what Friedrichs did .< ref > See, properties PI, PII, PIII and their consequence PIII < sub > 0 </ sub >.</ ref > This also clarifies why mollifiers are related to approximate identities.

However, as a consequence of the above properties, a Banach limit also satisfies:

* The Hilbert cube is compact, again a consequence of Tychonoff's theorem.

In this situation, the chain rule represents the fact that the derivative of is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula.

According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.

As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere ( complex plane and the point at infinity ) is constant.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.

An important consequence of the completeness theorem is that it is possible to recursively enumerate the semantic consequences of any effective first-order theory, by enumerating all the possible formal deductions from the axioms of the theory, and use this to produce an enumeration of their conclusions.

The compactness theorem says that if a formula φ is a logical consequence of a ( possibly infinite ) set of formulas Γ then it is a logical consequence of a finite subset of Γ.

This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.

Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem.

As a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial, the quotient of on division by.

An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.

A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic.

That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position ; this is a special case of Noether's theorem.

is a consequence of Stone's theorem on one-parameter unitary groups.

The embedding property is a consequence of a result known as Desargues ' theorem.

Quines are possible in any programming language that has the ability to output any computable string, as a direct consequence of Kleene's recursion theorem.

As a consequence of the noisy-channel coding theorem, noise limits the information content of an analog information carrier to be finite.

One consequence of Toda's theorem is that a polynomial-time machine with a # P oracle ( P < sup ># P </ sup >) can solve all problems in PH, the entire polynomial hierarchy.

As a consequence of the Weierstrass approximation theorem, one can show that the space C is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients ; there are only countably many polynomials with rational coefficients.