* In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. In the particular case of Hilbert spaces ( or more generally, inner product spaces ), an orthonormal basis is a set that is both complete and orthonormal.

A real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space R < sup > n </ sup > with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of R < sup > n </ sup >.

In this special case, the columns of U < sup >*</ sup > are eigenvectors of both A and B and form an orthonormal basis in C < sup > n </ sup >.

( This could be a " continuously indexed basis " or a mixture of continuous and discrete indexes, in which case we would have to use a rigged Hilbert space and be more careful about what we mean by orthonormal but that's an inessential detail for expository purposes.

In the special case of a manifold isometrically embedded into a higher dimensional Euclidean space, the covariant derivative can be viewed as the orthonormal projection of the Euclidean derivative along a tangent vector onto the manifold's tangent space.

This applies, for example, with a Hilbert space ; in which case a biorthogonal system reduces to an orthonormal system.

On the basis of the findings which led to this conclusion, the Court remanded the case to the District Court to determine the appropriate relief.

In the case of taxpaying corporate stockholders, the measure would be the lesser of the fair market value of the shares or Du Pont's tax basis for them, which is approximately $2.09 per share.

The determination of whether any particular condition is considered a disability is made on a case by case basis.

In the Georgacarakos v. Watts case Peter N. Georgacarakos filed a pro se civil-rights complaint in the United States District Court for the District of Colorado against 19 prison officials for " interference with the free exercise of his Ásatrú religion " and " discrimination on the basis of his being Ásatrú ".

Again a classic of Renaissance literature was the basis of the libretto by Boccherini, in this case a comic mock-epic by Tassoni, in which a war between Modena and Bologna ensues over a stolen bucket.

The coordinates of the vector are equal to the projections of the vector ( yellow ) onto the x-component basis vector ( green )-using the dot product ( a special case of an inner product, see below ).

Casuistry is the basis of case law in common law, and the standard form of reasoning applied in common law.

In the case of the CMM, for example, the basis for comparison would be the organizations ' software development processes.

In either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping.

A special case of this is a Lorentzian manifold, which is the mathematical basis of Einstein's general relativity theory of gravity.

To enter a Dutch doctoral defense, the candidate must hold a validated master degree ( a master degree of an acknowledged university, or a master equivalent degree validated on a case by case basis by the Dutch government ).

Diagnosis is done on a case by case basis, as lesions often affect surrounding cortex and deficits are not well conserved between patients.

Kaiser Permanente did not change its methods of evaluating whether or not new therapies were too " experimental " to be covered until it was successfully sued twice: once for delaying in vitro fertilization treatments for two years after the courts determined that scientific evidence of efficacy and safety had reached the " reasonable " stage ; and in another case where Kaiser refused to pay for liver transplantation in infants when it had already been shown to be effective in adults, on the basis that use in infants was still " experimental.

In nations where Islamic law is the basis of civil law, but has not been codified, as is the case of some Arab countries in the Middle East, fatwā by the national religious leadership are debated prior to being issued.

In nations where Islamic law is not the basis of law ( as is the case in various Asian and African countries ), different mujtahids can issue contradictory fatwā.

Through analysis of this and other of Hayek's works, Block purports, " in making the case against socialism, Hayek was led into making all sort of compromises with what otherwise appeared to be his own philosophical perspective – so much so, that if a system was erected on the basis of them, it would not differ too sharply from what this author explicitly opposed.

As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of ( special ) relativistic mechanics.

In the case where V is of finite dimension n it is common to choose a basis for V and identify GL ( V ) with GL ( n, K ) the group of n-by-n invertible matrices on the field K.

In diplomatic situations, governors-general are sometimes accorded a status akin to a head of state, but that is by tradition and on a case by case and person by person basis, not automatic.

Often this means making the central bank very aggressive about maintaining price stability, as was the case with the German Bundesbank or moving to some hard basis of currency such as a currency board.

Reloading may not be cost effective for occasional shooters, as it takes time to recoup the cost of the required equipment, but those who shoot on a regular basis will see benefit as the brass case ( the most expensive component ) can be reused many times ( with proper maintenance ).

This is the only case in modern history of a people of Britannic origin submitting without continued struggle to what they view as foreign domination.

he is very close to being a mental case.

What was only a vague suspicion in the case of Sherlock Holmes now appears as a direct accusation: the private eye is in danger of turning into his opposite.

In the case of social decay, form is displaced simply by the process of dissolution with no form at the terminus of the process.

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

This text from Dr. Huxley is sometimes used by enthusiasts to indicate that they have the permission of the scientists to press the case for a wonderful unfoldment of psychic powers in human beings.

There may be a case of this sort, but it is not one we wish to argue, here.

In any case but the last, such a course is sure to avenge itself upon the individual ; ;

The first thing to do is get her some money by a temporary but definite adjustment pending a final disposition of the case.

Such, he implies, is the case with his friend, who is not really a new convert himself but merely a favorer of new converts.

Although because of the important achievements of nineteenth century scholars in the field of textual criticism the advance is not so striking as it was in the case of archaeology and place-names, the editorial principles laid down by Stevenson in his great edition of Asser and in his Crawford Charters were a distinct improvement upon those of his predecessors and remain unimproved upon today.

That fact is very clearly illustrated in the case of the many present-day intellectuals who were Communists or near-Communists in their youth and are now so extremely conservative ( or reactionary, as many would say ) that they can define no important political conviction that does not seem so far from even a centrist position as to make the distinction between Mr. Nixon and Mr. Khrushchev for them hardly worth noting.

In any case, who ever thought that New York is typical of anything??

American history should clinch the case when Congress is asked to approve.

Senator Mundt's gross distortion of President Eisenhower's conversation into a denunciation of President Kennedy as too left wing, a statement Mr. Eisenhower declared to be entirely false, is another case in point.

The Connally amendment says that the United States, rather than the court, shall determine whether a matter is essentially within the domestic jurisdiction of the United States in a case before the World Court to which the United States is a party.

If the case is thus determined by us to be domestic, the court has no jurisdiction.

Besides the lack of an adequate ethical dimension to the Governor's case, one can ask seriously whether our lead over the Russians in quality and quantity of nuclear weapons is so slight as to make the tests absolutely necessary.

In my own case, I submit that such reasonable and fair exercise is woefully lacking.

In connection with any claim decided by the Commission pursuant to this Title in which an award is made, the Commission may, upon the written request of the claimant or any attorney heretofore or hereafter employed by such claimant, determine and apportion the just and reasonable attorney's fees for services rendered with respect to such claim, but the total amount of the fees so determined in any case shall not exceed 10 per centum of the total amount paid pursuant to the award.

in the case of a partnership or corporation, the existence of which has been terminated and on behalf of which an award is made, payment shall be made, except as provided in paragraphs ( 3 ) and ( 4 ), to the person or persons found by the Comptroller General of the United States to be entitled thereto ; ;