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The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky-Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by Augustin-Louis Cauchy. The corresponding inequality for integrals was published by Viktor Bunyakovsky and Hermann Schwarz. Schwarz gave the modern proof of the integral version.

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• The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky-Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by Augustin-Louis Cauchy. The corresponding inequality for integrals was published by Viktor Bunyakovsky and Hermann Schwarz. Schwarz gave the modern proof of the integral version. (en)
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• 38128 (xsd:integer)
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• 1040275385 (xsd:integer)
• Viktor Yakovlevich Bunyakovsky (en)
• Augustin-Louis Cauchy (en)
• Hermann Schwarz (en)
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• hidden (en)
dbp:first
• Hermann (en)
• Viktor (en)
• E. D. (en)
• Augustin-Louis (en)
dbp:id
• C/c020880 (en)
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• Schwarz (en)
• Solomentsev (en)
• Bunyakovsky (en)
• Cauchy (en)
dbp:left
• true (en)
dbp:mathStatement
• For reals : (en)
• For a 2-positive map between C*-algebras, for all in its domain, : : (en)
• Let and be arbitrary vectors in an inner product space over the scalar field where is the field of real numbers or complex numbers Then where in addition, equality holds in the if and only if and are linearly dependent. (en)
• If is a unital positive map, then for every normal element in its domain, we have and (en)
• If is a positive linear functional on a C*-algebra then for all (en)
dbp:name
• Callebaut's Inequality (en)
• Cauchy-Schwarz inequality (en)
• Kadison–Schwarz inequality (en)
• Cauchy–Schwarz inequality for positive functionals on C*-algebras (en)
dbp:note
• Modified Schwarz inequality for 2-positive maps (en)
• Named after Richard Kadison (en)
dbp:proof
• ; (en)
• A well-known way to write Cauchy-Schwarz is, for : : Now, to simplify, let : Thus, the statement we are trying to prove can be written as . This rearranges to , and if we have the quadratic equation , the discriminant is . Therefore, it will be sufficient to prove that this quadratic has no real roots , meaning: : Substituting back in our values of , we get: : Again, this rearranges to: : This factors to: : Which is true by the trivial inequality (en)
• The special case of was proven above so it is henceforth assumed that Let : It follows from the linearity of the inner product in its first argument that: : Therefore, is a vector orthogonal to the vector We can thus apply the Pythagorean theorem to : which gives : The Cauchy–Schwarz inequality follows by multiplying by and then taking the square root. Moreover, if the relation in the above expression is actually an equality, then and hence (en)
• the definition of then establishes a relation of linear dependence between and The converse was proved at the beginning of this section, so the proof is complete. ⯀ (en)
• thus which shows that and are linearly dependent. Since the converse was proved above, the proof of the theorem is complete. ⯀ Details of 's elementary expansion are now given for the interested reader. Let and so that and Then : Note that this expansion does not require to be non-zero; however, must be non-zero in order to divide both sides by and to deduce the Cauchy-Schwarz inequality from it. Swapping and gives rise to: : and thus : (en)
• The special case of was proven above so it is henceforth assumed that As is now shown, the Cauchy–Schwarz equality is an almost immediate corrollary of the following : which is readily verified by elementarily expanding and then simplifying. Observing that the left hand side of is non-negative proves that from which the follows . If then the RHS of is which is only possible if (en)
• The special case of was proven above so it is henceforth assumed that Let be defined by : Then : Therefore, or If the inequality holds as an equality, then and so thus and are linearly dependent. The converse was proved at the beginning of this section, so the proof is complete. ⯀ (en)
dbp:title
• Proof of the trivial parts: Case where a vector is and also one direction of the (en)
• Cauchy inequality (en)
• Proof 1 (en)
• Proof 2 (en)
• Proof 3 (en)
• Proof 4 (en)
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• yes (en)
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dbp:year
• 1821 (xsd:integer)
• 1859 (xsd:integer)
• 1888 (xsd:integer)
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• The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky-Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by Augustin-Louis Cauchy. The corresponding inequality for integrals was published by Viktor Bunyakovsky and Hermann Schwarz. Schwarz gave the modern proof of the integral version. (en)
rdfs:label
• Cauchy–Schwarz inequality (en)
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