- #1
Rederick
- 12
- 0
I expanded (x+y),(x+y) and got x^2+y^2 > 2xy then replaced 2xy with 2|x,y| but now I'm stuck.
I need to get it to ||x+y|| <= ||x|| + ||y||. Am I close?
I need to get it to ||x+y|| <= ||x|| + ||y||. Am I close?
The Minkowski Inequality is a mathematical inequality that states that for any two sequences of real numbers, the sum of their absolute values raised to a power p will be less than or equal to the sum of the absolute values of the products of the two sequences raised to the same power.
The Cauchy-Schwartz Inequality is a mathematical inequality that states that for any two vectors in an inner product space, the absolute value of their inner product is less than or equal to the product of their norms.
Cauchy-Schwartz Inequality can be used to prove Minkowski Inequality by recognizing that the inner product of two sequences in an inner product space is equivalent to the product of their norms. This allows us to apply Cauchy-Schwartz Inequality to the inner product, which then leads to the proof of Minkowski Inequality.
The steps to prove Minkowski Inequality using Cauchy-Schwartz Inequality are as follows:
Minkowski Inequality has various applications in mathematics, physics, and engineering. Some examples include its use in proving the triangle inequality in metric spaces, its application in the study of Fourier series, and its use in analyzing the stability of numerical methods in computational mathematics.