Answer to Question #139494 in Functional Analysis for anjali

Question #139494

Lex X be an inner product space over R. If x ,y X are such that ||x + y||=||x-y||then show that x  y.


1
Expert's answer
2020-10-21T16:52:46-0400

Let "\\langle \\cdot\\ , \\cdot\\rangle" be inner product on "X" .

Elements "x,y\\in X" are such that "\\|x+y\\|=\\|x-y\\|" .

If "\\|x+y\\|=\\|x-y\\|" , then "\\|x+y\\|^2=\\|x-y\\|^2" .

"\\|x+y\\|^2=\\langle x+y,x+y\\rangle=\\langle x,x\\rangle +\\langle x,y\\rangle+\\langle y,x\\rangle +\\langle y, y\\rangle =\\|x\\|^2+\\|y\\|^2+2\\langle x,y\\rangle"

"\\|x-y\\|^2=\\langle x-y,x-y\\rangle=\\langle x,x\\rangle -\\langle x,y\\rangle-\\langle y,x\\rangle +\\langle y, y\\rangle =\\|x\\|^2+\\|y\\|^2-2\\langle x,y\\rangle"

Then we have that "0=\\|x+y\\|^2-\\|x-y\\|^2=4\\langle x,y\\rangle" .

"\\langle x,y\\rangle =0" if and only if "x\\perp y" .


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