micromass said:What is <u,v> and ||u|| in [itex]\mathbb{R}^2[/itex]??
Can you rewrite the Cauchy-Schwarz inequality in [itex]\mathbb{R}^2[/itex]??
sam0617 said:<u,v> in R2 would be u1v1 + u2v2 I believe..
and ||u|| in R2 is √u2
micromass said:Good
No, that would be [itex]\sqrt{u_1^2+u_2^2}[/itex].
Anybody, do you recognize something like [itex]u_1v_1+u_2v_2[/itex] in your OP??
micromass said:Can you write out
[tex]<u,v>^2\leq \|u\|^2\|v\|^2[/tex]
??
Just write out what <u,v> and ||u|| mean...
The Cauchy-Schwarz inequality is a mathematical inequality that states that for any two vectors in an inner product space, the square of the dot product of the two vectors is always less than or equal to the product of the square of the norm of each vector. This inequality is commonly used in proving theorems in mathematics and is especially useful in the study of real numbers.
The Cauchy-Schwarz inequality can be used to prove that the dot product of two vectors is always less than or equal to the product of their norms. By setting the vectors to be the two sides of a triangle with sides a and b and the angle between them as theta, we can use the inequality to prove that the length of the third side (c) must be a real number, thus proving the existence of real values for a, b, and theta.
Yes, the Cauchy-Schwarz inequality can be used for any type of triangle, as long as the vectors used in the inequality are the two sides of the triangle and the angle between them. This is because the inequality is a general statement about the relationship between the dot product and the norms of vectors, and is not limited to any specific type of triangle.
Yes, the Cauchy-Schwarz inequality has many applications in mathematics, physics, and other sciences. It is commonly used in optimization problems, statistics, and probability theory. It also has applications in geometry, functional analysis, and linear algebra.
Yes, the Cauchy-Schwarz inequality is always true and is considered to be one of the fundamental inequalities in mathematics. It can be proven using various methods, including the Cauchy-Schwarz proof, the geometric proof, and the algebraic proof. Its validity is not limited to any specific set of numbers or vectors, making it a powerful tool in mathematical proofs and applications.