This is also called Cauchy Schwarz inequality, but requires for its statement that 2 } } and 2 } } are finite to make sure that the inner product of & thinsp; and is well defined.
32.
For instance, the Bunyakovsky inequality, the Wielandt inequality, and the Cauchy & ndash; Schwarz inequality are equivalent to the Kantorovich inequality and all of these are, in turn, special cases of the H�lder inequality.
33.
If the norm arises from an inner product ( as is the case for Euclidean spaces ), then the triangle inequality follows from the Cauchy Schwarz inequality as follows : Given vectors and, and denoting the inner product as:
34.
Among other things, Schwarz improved the proof of the Riemann mapping theorem, developed a special case of the Cauchy Schwarz inequality, and gave a proof that the ball has less surface area than any other body of equal volume.
35.
The continuity of " B " is even easier to see : simply apply the Cauchy-Schwarz inequality and the fact that the Sobolev norm is controlled by the " L " 2 norm of the gradient.
36.
Also in 1974 he made the first observation of sub-Poissonian statistics for light ( via a violation of the Cauchy Schwarz inequality for classical electromagnetic fields ), and thereby, for the first time, demonstrated an unambiguous particle-like character for photons.
37.
Writing the non-zero eigenvalues of G as \ lambda _ 1, \ ldots, \ lambda _ r with r \ leq n and applying the Cauchy-Schwarz inequality to the inner product of an r-vector of ones with a vector whose components are these eigenvalues yields
38.
Bunyakovsky was a mathematician, noted for his work in theoretical mechanics and number theory ( see : Bunyakovsky conjecture ), and is credited with an early discovery of the Cauchy Schwarz inequality, proving it for the infinite dimensional case in 1859, many years prior to Hermann Schwarz's works on the subject.
39.
In particular, if the set is normalized ( though not necessarily orthogonal ) then the diagonal elements will be identically 1 and the magnitude of the off-diagonal elements less than or equal to one with equality if and only if there is linear dependence in the basis set as per the Cauchy Schwarz inequality.
40.
Alternatively, it is possible to rewrite the equation of the plane using dot products with \ mathbf { p } in place of the original dot product with \ mathbf { v } ( because these two vectors are scalar multiples of each other ) after which the fact that \ mathbf { p } is the closest point becomes an immediate consequence of the Cauchy Schwarz inequality.
