# Tagged Questions

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### Apollonius’ Identity inner product space

$||z-x||^2+||z-y||^2=\frac{1}{2}||x-y||^2+2||z-\frac{x+y}{2}||^2$ I proved it by expanding both sides and i found both sides are equal. Are there any easy way to prove it?
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### An inequality for inner product space: $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$

In a inner product space show that the following inequality holds. $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$ I am stuck in proving this inequality
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### Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
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### Proving an inequality on $\sum_{1\leq i,j \leq n} \langle c_i ,c_j \rangle \times \langle l_i ,l_j \rangle$

This is a question that stumped me during an exam I took today. Let $c_1,...,c_n,l_1,...,l_n$ be vectors of $\mathbb R^n$ and $\langle .,.\rangle$ denote the dot product. Prove that ...
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### Proving the inequality of Cauchy-Schwarz in an Euclidean space. [duplicate]

It says let (G, <.,.>) be an euclidean space. Show that for all x, y belonging to G: modulus<x,y> <= sqrt<x,x> * sqrt<y,y> and in the ...