Prove that $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$ using those 4 axioms of inner product. From which axiom should I start?
2026-04-28 15:14:08.1777389248
prove that $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$
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Hint. Note that $$\|u+v\|^2+\|u-v\|^2=\langle u+v,u+v\rangle+\langle u-v,u-v\rangle\\=\langle u+v,u\rangle+\langle u+v,v\rangle +\langle u-v,u\rangle-\langle u-v,v\rangle.$$ where we used the linearity in the second argument. Can you take it from here?