Show that | $\left\lVert x \right\rVert - \left\lVert y \right\rVert$ | $\leq$ $\left\lVert x - y \right\rVert$ for any x and y in a normed space.

Here is my attempt:

Take the triangle inequality \begin{align*} \left\lVert x + y \right\rVert &\leq \left\lVert x + y \right\rVert + \left\lVert y \right\rVert \\ &= \left\lVert x - y + y \right\rVert + \left\lVert y \right\rVert \\ &\leq \left\lVert x - y\right\rVert + \left\lVert y \right\rVert + \left\lVert y \right\rVert. \end{align*}

Then we move the two $\left\lVert y \right\rVert$'s to the other side \begin{align*} \left\lVert x-y \right\rVert &\geq \left\lVert x + y \right\rVert - \left\lVert y \right\rVert - \left\lVert y \right\rVert \\ &\geq \left\lVert x \right\rVert + \left\lVert y \right\rVert - \left\lVert y \right\rVert - \left\lVert y \right\rVert\\ &\geq \left\lVert x \right\rVert - \left\lVert y \right\rVert.\end{align*}

I am not sure if I can separate the $\left\lVert x + y \right\rVert$ into $\left\lVert x \right\rVert + \left\lVert y \right\rVert$ like I did and I am not sure how to get absolute values so any hints would be great.

  • $\begingroup$ You can not separate the $||x+y||$ like you did. Triangle inequality gives $||x+y||\leq||x||+||y||$, not $\geq$. To solve the exercise use the reverse triangle inequality: $||x||\leq||x-y||+||y||$, so $||x-y||\geq||x||-||y||$. $\endgroup$ – SmileyCraft Oct 17 '18 at 23:27
  • $\begingroup$ In the textbook I am using this is called the triangle inequality so I'm not sure I am aloud to do that. $\endgroup$ – Polectio Oct 17 '18 at 23:30
  • $\begingroup$ You can prove the reverse triangle inequality yourself. It is only a few deductions. $\endgroup$ – SmileyCraft Oct 17 '18 at 23:32

Assume that $\| x\|\geq \|y\|$. Then $$|\|x\|-\|y\|| =\|x\|-\|y\|\leq \|x-y\|$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.