If $X$ is a normed space and $Y$ is a finite dimensional subspace, then there exists a continuous linear projection $P$ from $X$ to $Y$. Our teacher gave us the instruction to use the following fact: Let $x_1,\cdots,x_n$ in $X$ be linearly independent. Then there exist $x'_1,\cdots,x'_n$ in $X'$ such that $x'_k(x_r)= \delta_{kr}, 1 \le k,r \le n$. How does one proceed with this assumption? Thank you!
continuous projections to finite dimensional subspaces of normed spaces
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normed-spaces
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0Hello Yuki, please do help me with how one is supposed to use the property to solve the question. Why is it possible to define the mapping P like this? Thank you. – 2012-06-10
1 Answers
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@Yuki gave you the answer.
Define $P:X\to X$ by $Px = \sum x_k'(x) x_k$. Show that $P$ is linear, continuous and $P P = P$.
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0$P(Px) = \sum_k x_k^'(Px) x_k = \sum_k x_k^'(\sum_j x_j^'(x) x_j) x_k = \sum_k \sum_j x_j^'(x) x_k^'( x_j) x_k = \sum_k x_k^'(x) x_k = Px$ – 2012-06-10