[問題]線代問題...希望高手可以幫忙..謝謝

[問題]線代問題...希望高手可以幫忙..謝謝

cloudsea 於 星期四 四月 20, 2006 7:56 pm


Let ( V,<,>)be a finite-dimensional inner product space over C and F : V-->C  be linear .  Show that there exists an unique y∈V such that F(x) = <x,y> for all x∈ V             沒什麼想法......想令V為R^n..但又不能這樣想..因為這樣好像只是證其中一特例..

cloudsea
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註冊時間: 2005-07-05

Trivial 於 星期日 八月 13, 2006 11:20 pm


This is a basic fact of inner product space.
Take an orthonormal basis of V, (e_1,...,e_n)
Let F(e_i) = a_i, and take y = a_1e_1+...+a_ne_n
By checking,
F(e_i) = <e_i, y> because the basis is orthonormal
so we are done

Trivial
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