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Section D.3 Exercises
-
Show directly from the definition that for
\(x=(x_1,\dots,x_n)\in\R^n\text{,}\) one has
\(\|x\|=0\) if and only if
\(x=0\text{.}\)
-
Let
\(x,y\in\R^n\text{.}\) Show that if
\(x\cdot y=0\text{,}\) then
\(\|x+y\|^2=\|x\|^2+\|y\|^2\text{.}\)
-
Prove that for all
\(x,y\in\R^n\text{,}\) \(\bigl|\,\|x\|-\|y\|\,\bigr|\le \|x-y\| \le \|x\|+\|y\|\text{.}\)
-
For
\(x,y\in\R\text{,}\) the Euclidean distance on
\(\R\) is
\(d(x,y)=|x-y|\text{.}\) Verify the four metric axioms in this case.
