soit H espace de Hilbert montrez que: d(x,{a})^⌊ = ((∣<x,a>∣)/(∣∣a∣∣))


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Question Number 205607 by SANOGO last updated on 25/Mar/24
$soit H espace de Hilbert montrez que: d(x,{a})^⌊ = ((∣<x,a>∣)/(∣∣a∣∣))$
$s o i t H e s p a c e d e H i l b e r t$ $m o n t r e z q u e :$ $d {(x, {a})}^{⌊} = \frac{∣< x, a >∣}{∣∣ a ∣∣}$
	Answered by Berbere last updated on 25/Mar/24
	$(a/(∣a∣)),a_2 .....a_n une base orthonmal x=Σ_(i=1) ^n <x,(a_i /(∣a∣))>(a_i /(∣a∣)) x=<x,(a/(∣a∣))>(a/(∣a∣))+y;y∈{a}^⊥ d(x,{a})∣<x,(a/(∣a∣))>∣=∣(1/(∣a∣))<x,a>∣=((<x,a>)/(∣∣a∣∣))$
	$\frac{a}{∣ a ∣}, a_{2} . . . . . a_{n} u n e b a s e o r t h o n m a l$ $x = \underset{i = 1}{\sum^{n}} < x, \frac{a_{i}}{∣ a ∣} > \frac{a_{i}}{∣ a ∣}$ $x =< x, \frac{a}{∣ a ∣} > \frac{a}{∣ a ∣} + y; y \in {a}^{⊥}$ $d (x, {a}) ∣< x, \frac{a}{∣ a ∣} >∣=∣ \frac{1}{∣ a ∣} < x, a >∣= \frac{< x, a >}{∣∣ a ∣∣}$
	Commented by SANOGO last updated on 25/Mar/24

	$t h a n k y o u$
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