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damabo

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## Homework Statement

show that (U[itex]\bot[/itex])[itex]\bot[/itex]=U, if (ℝ,V,+,[,.,]) an Euclidian space en U is a linear subspace of V.

## Homework Equations

## The Attempt at a Solution

suppose [itex]\beta={u_1,...,u_k}[/itex] is an orthonormal basis of U.

pick u in U. Then [itex]u=x_1u_1+...+x_ku_k[/itex] for certain x_1,...,x_k in ℝ.

pick u'_1,...,u'_k as orthonormal basisvectors of U[itex]\bot[/itex], where u'_i[itex]\bot[/itex]u_i for all i. and consequently take basisvectors u''_1,...,u''_k for U[itex]\bot\bot[/itex] that are orthogonal to the previous. then it follows that u''=x_1u''_1+...+x_ku''_k = x_1u_1+...+x_ku_k = u.

this means that U[itex]\subset[/itex]U[itex]\bot\bot[/itex] and also that both have the same dimensions, and consequently that they are the same subspaces (since [itex]span(u_1,...,u_k)=span(u''_1,...,u''_k)=ℝ^k [/itex]).

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