## How do you prove Cauchy inequality?

As explained in class, if you believe that vectors in hundreds of dimensions act like the vectors you know and love in R2, then the Cauchy-Schwartz inequality is a consequence of the law of cosines. Specifically, u · v = |u||v|cosθ, and cosθ ≤ 1.

**What is the Cauchy-Schwarz inequality used for?**

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.

### What condition is needed for equality to hold in the Cauchy-Schwarz inequality?

Thus the Cauchy-Schwarz inequality is an equality if and only if u is a scalar multiple of v or v is a scalar multiple of u (or both; the phrasing has been chosen to cover cases in which either u or v equals 0).

**Which of the following is the Cauchy-Schwarz inequality?**

( ∑ i = 1 n a i 2 ) ( ∑ i = 1 n b i 2 ) ≥ ( ∑ i = 1 n a i b i ) 2 . Not only is this inequality useful for proving Olympiad inequality problems, it is also used in multiple branches of mathematics, like linear algebra, probability theory and mathematical analysis. …

## How do you prove triangle inequalities?

Triangle Inequality Proof

- Since the sum of any two sides is greater than the third, then the difference of any two sides will be less than the third.
- The sum of any two sides must be greater than the third side.
- The side opposite to a larger angle is the longest side in the triangle.

**What is Cauchy-Schwarz inequality example?**

Example question: use the Cauchy-Schwarz inequality to find the maximum of x + 2y + 3z, given that x2 + y2 + z2 = 1. We know that: (x + 2y + 3x)2 ≤ (12 + 22 32)(x2 + y2 + z2) = 14. Therefore: x + 2y + 3z ≤ √14.

### Which one of the following is triangle inequality?

Triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle is greater than or equal to the third side; in symbols, a + b ≥ c. In essence, the theorem states that the shortest distance between two points is a straight line.

**Does Cauchy-Schwarz hold for complex numbers?**

The Cauchy-Schwarz inequality remains true but the proof must be modified slightly because now the scalar t in our proof might need to be complex. In this case just use 0 ≤ |Z − tW|2 where t = 〈Z, W〉/|W|2.

## What is the difference between the three theorems under triangle inequality?

Hence, let us check if the sum of two sides is greater than the third side. Therefore, the sides of the triangle do not satisfy the inequality theorem….Q. 3: If the two sides of a triangle are 2 and 7. Find all the possible lengths of the third side.

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**How do you prove triangles inequalities?**