Cauchy–Schwarz inequality
The Cauchy- Schwarz inequality, also known as the Schwarz inequality and Cauchy - Schwarz inequality Bunjakowski, is an inequality that is used in many areas of mathematics, such as linear algebra ( vectors) in the analysis ( infinite series ), in probability theory and of integration of products. In quantum mechanics also plays an important role, such as in the proof of the Heisenberg uncertainty principle.
- 5.2.1 real case
- 5.2.2 Complex case
- 6.1 Evidence for the real case
- 6.2 Conditions for equality
General case
The inequality states that if and elements of a real or complex vector space with inner product, then apply for the scalar or inner product of the relationship
Equality holds if and only if and are linearly dependent.
Equivalent formulations but using the induced by the inner product standard:
Or
In the real case, one can dispense with the amount of strokes:
Special cases
In the space used with the standard scalar product, we obtain:
In the case of square-integrable complex-valued functions we obtain:
For square-integrable random variables is obtained:
These three inequalities are generalized by the Hölder inequality.
On square matrices applied, one obtains for the track:
In can be the statement of the Cauchy- Schwarz inequality in the form of an equation specification:
The summand is always non- negative. He is exactly zero when and are linearly dependent.
History
It is named after Augustin Louis Cauchy inequality, Viktor Yakovlevich Bunjakowski and Hermann Amandus Schwarz. When we find the Cauchy sum form of inequality in its analysis algébrique (1821 ). The integral form of the inequality was historically the first time in 1859 posted by Bunjakowski in a paper on inequalities between integrals; Black published his work until 50 years later.
Applications
In a vector space with inner product can be calculated from the Cauchy- Schwarz inequality, the triangle inequality for the induced norm
Derived, and thus show subsequently that such a defined standard fulfills the standard axioms.
A further consequence of the Cauchy-Schwarz inequality, in that the inner product is a continuous function.
The Cauchy- Schwarz inequality ensures that in the expression
The amount of the fraction is always less than or equal to one, ie, so that is well-defined and thus the angle can be generalized to any space with inner product.
In physics, the Cauchy- Schwarz inequality is used in the derivation of Heisenberg's uncertainty principle.
Proof of the inequality
If one of the vectors is the zero vector, then the Cauchy- Schwarz inequality is trivially satisfied. In the following proofs, therefore, is partly without notice and provided.
Special case of real standard scalar
Proof of the inequality of the arithmetic and geometric means
A proof of the Cauchy-Schwarz 's inequality may be made from the arithmetic and geometric center, for example, using the relationship:
You defined for the values
It follows from the inequality of the arithmetic and geometric means, the relationship
This is immediately followed by the Cauchy- Schwarz inequality.
Evidence from the rearrangement inequality
Another proof of the Cauchy- Schwarz inequality results from the rearrangement inequality. Substituting
And thus applies
Because of the rearrangement inequality is now
In summary, we obtain therefore
Hence the Cauchy- Schwarz inequality.
General scalar
The evidence given above prove only the special case of the Cauchy- Schwarz inequality for the standard scalar product in. The proof for the general case of the scalar product in a vector space with inner product, however, is simple.
Real case
Applies provided. For each
If we choose now specifically we obtain
So
Taking the square root yields now exactly the Cauchy- Schwarz inequality
Complex case
The proof in the complex case is similar, however, note that the scalar product of any bilinear form, but a Hermitian form in this case. The proof is carried out for the variant linear in the first and semilinear in the second argument; the reverse variant is selected, then take the appropriate places the complex conjugate.
If so, the statement is clear. Be. For each
Here now leads the special election on
So
Generalization for positive semidefinite symmetric bilinear forms
One can reformulate the proof of the theorem so that the positive definiteness of the scalar product is not used. Thus, the statement also applies to any bilinear positive semidefinite, symmetric (or Hermitian sesquilinear ).
Proof for the real case
We choose the same approach as in the proof that uses the scalar product, meets here but the choice
Similarly to the above proof you infer
And the assertion is shown when converges to 0.
Conditions for equality
Again, the situation is also conceivable that the inequality is an equality, such as when are linearly dependent ( as the scalar product ). However, cases are also conceivable, where the equality occurs, without having a linear relationship exists. Take, for instance, a degenerate bilinear form. Then there exists an such that for all of the vector space. Now let any of the vector space. This gives then
And
So
Even in the case that linear and independent.