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The norm is a continuous function on its vector space, but I a little unsure of how to interpret this. Does it mean that if we are in e.g. a Hilbert space with an orthonormal basise_{i}(iis a positive integer), then we have

[tex]

\left\| {\mathop {\lim }\limits_{N \to \infty } \sum\limits_{i = 1}^N {x_i e_i } } \right\| =\mathop {\lim }\limits_{N \to \infty } \left\| {\sum\limits_{i = 1}^N {x_i e_i } } \right\|

[/tex]

for some vector x = Σ x_{i}e_{i}in the Hilbert space? Or does the above operation come from the continuity of taking limits?

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# Norms in vector spaces

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