Can anyone give a simple example for it? [![enter image description here][1]][1]
[1]: https://i.sstatic.net/d3KiB.jpg . Please expalin the example shown in pic.
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2But you've already got the example marked in yellow in your image?! – joriki Jul 01 '18 at 05:17
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Possible duplicate of An example of a norm which can't be generated by an inner product – Martin Sleziak Jul 04 '18 at 05:51
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What you have written down is that a normed linear space is an inner product space if and only if the parallelogram law holds. For a simple example of a normed spaced that is not an inner product space, consider the space of real-valued continuous functions on $[0,1],$ with norm $$\|f\|=\sup_{x\in[0,1]}|f(x)|$$ It is easy to see that this is a real vector space and to verify that $\|\cdot\|$ is a norm.
To see that it is not an inner product space we just need one example where the parallelogram law fails. Take $f(x)=x,\ g(x)=x^2.$ Then $\|f\|=\|g\|=1,\ \|f+g\|=2,\ \|f-g\|=1/4,$ and one can check that the parallelogram law does not hold.
saulspatz
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