Example 1.2. a) Let X = R and take d(x, y) = |x − y|. This is the most basic and important example. b) More generally, let N ≥ 1, let X = R , and take d(x, y) = ||x − y|| = √∑N i=1(xi − yi). It is very well known but not very obvious that d satisfies the triangle inequality. This is a special case of Minkowski’s Inequality, which will be studied later. c) More generally let p ∈ [1,∞), let N ≥ 1...

where the first inequality is an application of the triangle inequality, the second follows from (1) and (2), and the third from the choice of m. Therefore, since our choice of > 0 was arbitrary, we conclude that the subsequence (xnm)→ b. A similar argument using the sequence (zm) given by zm = inf{xm, xm+1, . . .} and the version of Lemma 1.3.7 suitable for infima (see Exercise 1.3.2, which yo...

Let C be a plane convex body. For arbitrary points , denote by , n a b E  ab the Euclidean length of the line-segment . Let be a longest chord of C parallel to the line-segment . The relative distance between the points and is the ratio of the Euclidean distance between and b to the half of the Euclidean distance between and . In this note we prove the triangle inequality in with the relative ...

In this paper the entropy exchange for channels and states in infinite-dimensional systems are defined and studied. It is shown that, this entropy exchange depends only on the given channel and the state. An explicit expression of the entropy exchange in terms of the state and the channel is proposed. The generalized Klein's inequality, the subadditivity and the triangle inequality about the en...

Refining some results of S. S. Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if a is a unit vector in a real or complex inner product space (H; 〈., .〉), r, s > 0, p ∈ (0, s],D = {x ∈ H, ‖rx− sa‖ ≤ p}, x1, x2 ∈ D − {0} and αr,s ...

is not trivial to prove. In the one-dimensional case, the triangle inequality is an inequality on absolute values, and can be proven case-by-case. In R, it is best to use the following set-up. The usual inner product (or dot-product) on R is x · y = 〈x, y〉 = 〈(x1, . . . , xn), (y1, . . . , yn)〉 = x1y1 + . . .+ xnyn and |x| = 〈x, x〉. Context distinguishes the norm |x| of x ∈ R from the usual abs...

In this paper we study vp(n!), the greatest power of prime p in factorization of n!. We find some lower and upper bounds for vp(n!), and we show that vp(n!) = n p−1 + O(lnn). By using the afore mentioned bounds, we study the equation vp(n!) = v for a fixed positive integer v. Also, we study the triangle inequality about vp(n!), and show that the inequality pp > qq holds for primes p < q and suf...

Refining some results of Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if a is a unit vector in a real or complex inner product space (H ;〈·,·〉), r,s > 0, p ∈ (0,s], D = {x ∈ H ,‖rx− sa‖ ≤ p}, x1,x2 ∈D−{0}, and αr,s = min{(r2‖x...