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Let $V$ be an inner product space over $\mathbb{F}$.

If one defines $\lVert \bullet \rVert$ as $\sqrt{\langle \bullet, \bullet \rangle}$, then $\lVert \bullet \rVert$ is a norm on $V$.

However, if $\lVert \bullet \rVert$ is defined as ${\langle \bullet, \bullet \rangle}^{1/4}$, this is another norm.

When it is an inner product space, does $\lVert \bullet \rVert$ simply mean 2-norm if there's nothing additionally mentioned? and What is the notation for this $p$-norm on an inner product space?

Plus, I hope someone please explain me this briefly relating $L^p$ space that why do we need these various different norms.

Thank you in advance!

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Of course i checked wikipedia, but it just shows various types of norms, not motivation. – Jj- May 10 '13 at 9:55
How is $\langle \bullet, \bullet \rangle^{1/4}$ a norm? Did you check homogeneity? – Martin May 10 '13 at 10:18

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