Thus, we get (1′) easily from (3), by setting $A=y$, $B=x-y$. Reverse Triangle Inequality Theorem Problem: Prove the Reverse Triangle Inequality Theorem. Privacy policy. \end{equation} =& Also, … Then kv wk kvkk wk for all v;w 2V. -|x-y| \leq |x|-|y| \leq |x-y|. \end{equation} =&|x-y|.\nonumber We could handle the proof very much like a proof of equality. Solution: By the Triangle Inequality, |x−y| = |(x−a)+(a−y)|≤|x−a|+|a−y|≤ + =2 Thus |x−y| < 2 . Suppose |x−a| <, |y −a| <. which when rearranged gives \end{align}. Reverse (or inverse) triangle inequalities: ka+ bk 2 kak 2 k bk 2 ka+ bk 2 kbk 2 k ak 2 878O (Spring 2015) Introduction to linear algebra January 26, 2017 4 / 22 \end{equation}, $\left| |x|-|y| \right|^2 – |x-y|^2 = \left( |x| – |y| \right)^2 – (x-y)^2 = |x|^2 – 2|x| \cdot |y| +y^2 – x^2 + 2xy-y^2 = 2 (xy-|xy|) \le 0 \Rightarrow \left| |x|-|y| \right| \le |x-y|.$. Reverse triangle inequality. \begin{equation} I’ve seen this proof, however it’s too advanced for me as it involves metric spaces – I’d like a simple proof using the known and simple triangle inequality I wrote in the question, thanks. In the case of a norm vector space, the statement is: The proof for the reverse triangle uses the regular triangle inequality, and. « Find the area of a parallelogram using diagonals. We will now look at a very important theorem known as the triangle inequality for inner product spaces. |x-y|=-x+y=-(x-y),&y\geq{}x\geq0\\ We study reverse triangle inequalities for Riesz potentials and their connection with polarization. The reverse triangle inequality is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. $$, On the other hand, the known triangle inequality tells us that “the sum of the absolute values is greater than or equal to the absolute value of the sum”: $$ The truly interested reader can find full proofs in Professor Bhatia’s notes (follow the link above) or in [1]. The difficult case The proof of the triangle inequality follows the same form as in that case. Triangle inequality Lemma (Triangle inequality) Given a;b 2RN, ka+ bk 2 kak 2 + kbk 2: Proof uses Cauchy-Schwarz inequality (do on board) When does this inequality hold with equality? Problem 6. \end{equation*} A vector v 2V is called a unit vector if kvk= 1. $$||x|-|y||=||x-y+y|-|y||\le ||x-y|+|y|-|y||=||x-y||=|x-y|.$$, Explicitly, we have | x − y | ≥ | x | − | y |. Reverse Triangle Inequality Proof Please Subscribe here, thank you!!! proofwiki.org/wiki/Reverse_Triangle_Inequality. For all $x,y\in \mathbb{R}$, the triangle inequality gives \blacksquare In a normed vector space V, one of the defining properties of the norm is the triangle inequality: Proposition 1 Reverse Triangle Inequality Let V be a normed vector space. 1 Young’s inequality: If p,q > 1are such that 1 p + 1 q =1, then xy ≤ xp p + yq q. Combining these two facts together, we get the reverse triangle inequality: | x − y | ≥ | | x | − | y | |. Proof. What is the main concepts going on in this proof? (Otherwise we just interchange the roles of x and y.) Let y ≥ 0be ﬁxed and consider the function $$ Theorem The area of a triangle with given perimeter 2p = a+b+c is maximum if the sides a, b, c are equal. De nition: Unit Vector Let V be a normed vector space. \end{equation*} |x|-|y|\ge -|x-y|\;.\tag{2} We don’t, in general, have $x+(x-y)=y$. For first and second triangle inequality, Combining these two statements gives: Proof of the corollary: By the first part, . |x+y|\le|x|+|y|. \end{equation} Since the real numbers are complex numbers, the inequality (1) and its proof are valid also for all real numbers; however the inequality may be simplified to |x|+|x-y|\ge |y|\tag{2′} The proof of the triangle inequality is virtually identical. asp.net – How to use C# 6 with Web Site project type? We can write the proof in a way that reveals how we can think about this problem. Furthermore, (1) and (2) can be written in such a form easily: For plane geometry, the statement is: [19] Any side of a triangle is greater than the difference between the other two sides . Interchaning $x\leftrightarrow y$ gives $$ The validity of the reverse triangle inequality in X,i.e. Therefore, what we need to prove are (both of) the following: using case 1) x;y 0, and case 2) x 0, y … Hence: Triangle inequality giv es an upp er bound 2 − , whereas reverse triangle ine qualities give lower bounds 2 − 2 √ 2 for general quantum states and 2 − 2 for classical (or commuting) \begin{equation*} \end{equation} Recall that one of the defining properties of a matrix norm is that it satisfies the triangle inequality: So what can we say about generalizing the backward triangle inequality to matrices? \begin{equation} https://goo.gl/JQ8Nys Reverse Triangle Inequality Proof. |A|+|B|\ge |A+B|\;\tag{3} \begin{array}{ll} Combining these two facts together, we get the reverse triangle inequality: WLOG, consider $|x|\ge |y|$. |-x+y|=x+y\leq{}x-y,&x\geq-y\geq0 Use the triangle inequality to see that 0 ja bj= ja a n+ a n bj ja a nj+ ja n bj. $$ |x|-|y|\leq |x-y| \tag{1}. |-x-y|=x+y\leq-x+y=-(x-y),&y\geq-x\geq0\\ PROOF By the triangle inequality, kvk= k(v w) + wk kv wk+ kwk; and the desired conclusion follows. 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