dispiriton said:Homework Statement
I am told to try and solve <x - ty, x - ty> where t = <x,y>/<y,y>
However, I am stuck at that equation and unable to manipulate it to get rid of the *
dispiriton said:<x - ty, x - ty> = <x,x> - <x,ty> - <ty,x> + <ty,ty>
= mod(x)^2 + mod(y)^2 (t^2) - t*<x,y> - (<x,ty>)*
Undoubtedly0 said:Greetings dispiriton! What do you mean by "the *"?
The Cauchy Schwarz inequality for complex numbers states that for any two complex number sequences, the inner product of the sequences is less than or equal to the product of the norms of the sequences.
The proof of the Cauchy Schwarz inequality for complex numbers involves using the properties of inner products and the Cauchy Schwarz inequality for real numbers. The complex numbers are treated as vectors in a real vector space, and the proof follows a similar logic as the proof for real numbers.
Yes, the Cauchy Schwarz inequality can be extended to higher dimensions, including infinite dimensions. This is known as the generalized Cauchy Schwarz inequality.
The Cauchy Schwarz inequality has many applications in mathematics and physics, including in the study of Fourier series, differential equations, and quantum mechanics. It is also used in optimization problems and in proving other important mathematical theorems.
No, the Cauchy Schwarz inequality is applicable to a wide range of mathematical objects, including real numbers, vectors, matrices, and functions. It is a fundamental inequality in mathematics with many applications in different fields.