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#### sweatingbear

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- May 3, 2013

- 91

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- May 3, 2013

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- Jan 26, 2012

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$$ \| \, f \|_{1}:=\int_{X} \left| \, f \right| \, d\mu.$$

It might be confusing if you used single bars on the LHS of this definition.

- Feb 15, 2012

- 1,967

\(\displaystyle \|\alpha v\| = |\alpha|\cdot \|v\|\)

where it makes sense to distinguish between the two types of "norms" being used on the vector space, and the underlying field.

Nevertheless, in many abstract treatments of linear algebra, only single vertical bars are used, with the double-vertical bar used for linear algebra with physical interpretations (where vectors and scalars represent different KINDS of entities).

In terms of complex numbers, the complex modulus turns out to BE the absolute value on the real line...the trouble is, in \(\displaystyle \Bbb R^n\) there's no "natural" line through the origin to pick as "the real line" (there are certain exceptions for the special cases n = 1, 2, 4, 8 and 16, but these are too complicated to go into here).

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- May 3, 2013

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Fair enough, thanks!