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Normed Vector Space

Consider the vector space R^2 with the norm ║(x,y)║ = │x │+│y │
Show that the set
U = { u element of R^2 : 0< &#9553;u&#9553; < 1}
is an open set in this normed vector space.

Solution Preview

Proof:
For any u=(x,y) in U, 0<&#9553;u&#9553;= |x|+|y| <1.
Then we can find a very small positive number e>0, such that
&#9553;(x+e,y+e)&#9553;=|x+e|+|y+e|<1,
i.e., ...

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