What would be an example of a not (topologically) closed subspace of a normed space?
Consider the space of continuous functions f:[0,1]->R with the supremum norm
[itex]\Vert f\Vert=\sup |f(x)|[/itex]. This is a normed vector space (in fact, a Banach space). The subspace of differentiable functions is not closed.
That's not a linear subspace though.
the linear span of a complete orthonormal set in hilbert space. it is dense, since all vectors are infinite series expansions of the, but not closed since not all vecors are finite linear combinations.
i.e. a hilbert basis is an o.n. set whose span is dense.
Or the set of indefinitely differentiable functions with compact support defined on R as a subset of L^p(R). It is a proper subspace and it is dense, therefor it is not closed.
I get the idea, thanks!
why the space of diffrental function not closed help me pleas quakly
Because you can find an example of a sequence of differentiable functions that converge uniformly to a non-differentiable function.
Simply take the space X of integrable functions on [0,1], equipped with the L_1 norm, and consider the subspace Y of continuous functions on [0,1]: one can find a Cauchy sequence of functions in Y whose limit is integrable but discontinuous, and is hence no longer in Y.
pleas give me eaxample
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