Get Cauchy%E2%80%93Schwarz Inequality essential facts below. View Videos or join the Cauchy%E2%80%93Schwarz Inequality discussion. Add Cauchy%E2%80%93Schwarz Inequality to your PopFlock.com topic list for future reference or share this resource on social media.
Moreover, the two sides are equal if and only if and are linearly dependent (meaning they are parallel: one of the vector's magnitudes is zero, or one is a scalar multiple of the other).
If and , and the inner product is the standard complex inner product, then the inequality may be restated more explicitly as follows (where the bar notation is used for complex conjugation): for , we have
Proof 1 —
Let and be arbitrary vectors in a vector space over with an inner product, where is the field of real or complex numbers. We prove the inequality
and that equality holds if and only if either or is a multiple of the other (which includes the special case that either is the zero vector).
If , it is clear that there is equality, and in this case and are also linearly dependent, regardless of , so the theorem is true.
Similarly if . One henceforth assumes that is nonzero.
Then, by linearity of the inner product in its first argument, one has
Therefore, is a vector orthogonal to the vector (Indeed, is the projection of onto the plane orthogonal to .)
We can thus apply the Pythagorean theorem to
and, after multiplication by and taking square root, we get the Cauchy-Schwarz inequality.
Moreover, if the relation in the above expression is actually an equality, then and hence ; the definition of then establishes a relation of linear dependence between and .
On the other hand, if and are linearly dependent, then there exists such that (since ). Then
This establishes the theorem.
Proof 2 —
Let and be arbitrary vectors in an inner product space over .
In the special case the theorem is trivially true.
Now assume that . Let be given by , then
Therefore, , or .
If the inequality holds as an equality, then , and so , thus and are linearly dependent.
On the other hand, if and are linearly dependent, then , as shown in the first proof.
There are many different proofs of the Cauchy-Schwarz inequality other than the above two examples.
When consulting other sources, there are often two sources of confusion. First, some authors define , to be linear in the second argument rather than the first.
Second, some proofs are only valid when the field is and not .
Titu's lemma (named after Titu Andreescu, also known as T2 lemma, Engel's form, or Sedrakyan's inequality) states that for positive reals, one has
It is a direct consequence of the Cauchy-Schwarz inequality, obtained upon substituting and This form is especially helpful when the inequality involves fractions where the numerator is a perfect square.
R2 (ordinary two-dimensional space)
Cauchy-Schwarz inequality in a unit circle of the Euclidean plane
In the usual 2-dimensional space with the dot product, let and . The Cauchy-Schwarz inequality is that
The form above is perhaps the easiest in which to understand the inequality, since the square of the cosine can be at most 1, which occurs when the vectors are in the same or opposite directions. It can also be restated in terms of the vector coordinates and as
where equality holds if and only if the vector is in the same or opposite direction as the vector or if one of them is the zero vector.
Rn (n-dimensional Euclidean space)
In Euclidean space with the standard inner product, the Cauchy-Schwarz inequality is
The Cauchy-Schwarz inequality can be proved using only ideas from elementary algebra in this case.
Consider the following quadratic polynomial in
Since it is nonnegative, it has at most one real root for , hence its discriminant is less than or equal to zero. That is,
Taking square roots gives the triangle inequality.
The Cauchy-Schwarz inequality 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 allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining:
The Cauchy-Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval [−1, 1] and justifies the notion that (real) Hilbert spaces are simply generalizations of the Euclidean space. It can also be used to define an angle in complexinner-product spaces, by taking the absolute value or the real part of the right-hand side, as is done when extracting a metric from quantum fidelity.
An inner product can be used to define a positive linear functional. For example, given a Hilbert space being a finite measure, the standard inner product gives rise to a positive functional by . Conversely, every positive linear functional on can be used to define an inner product , where is the pointwisecomplex conjugate of . In this language, the Cauchy-Schwarz inequality becomes
which extends verbatim to positive functionals on C*-algebras:
Theorem (Cauchy-Schwarz inequality for positive functionals on C*-algebras): If is a positive linear functional on a C*-algebra then for all , .
The next two theorems are further examples in operator algebra.
^Hassani, Sadri (1999). Mathematical Physics: A Modern Introduction to Its Foundations. Springer. p. 29. ISBN0-387-98579-4. Equality holds iff <c|c>=0 or |c>=0. From the definition of |c>, we conclude that |a> and |b> must be proportional.
Aldaz, J. M.; Barza, S.; Fujii, M.; Moslehian, M. S. (2015), "Advances in Operator Cauchy--Schwarz inequalities and their reverses", Annals of Functional Analysis, 6 (3): 275-295, doi:10.15352/afa/06-3-20
Cauchy, A.-L. (1821), "Sur les formules qui résultent de l'emploie du signe et sur > ou <, et sur les moyennes entre plusieurs quantités", Cours d'Analyse, 1er Partie: Analyse Algébrique 1821; OEuvres Ser.2 III 373-377