- #1
rhobymic
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Homework Statement
Let V be a complex inner product space and let S be a subset of V. Suppose that v in V is a vector for which
<s,v > + <v,s> [itex]\leq[/itex] <s,s>
Prove that v is in the orthogonal set S[itex]\bot[/itex]
Homework Equations
We have the three inner product relations:
1) conjugate symmetry
<x,y> = [itex]\overline{<y,x>}[/itex]
2) linearity
<x+y,z> = <x,z>+<y,z>
3) Def of a norm
||x|| = √<x,x>
There may be more that apply such as triangle inequality or the Cauchy–Schwarz inequality
but I am not sure
The Attempt at a Solution
I know that if v is in the set S[itex]\bot[/itex]
then s is orthogonal to v so <s,v> = <v,s> = 0
Therefor I am guessing through all these equations and the given inequality it can be shown that <s,v>+<v,s> needs to be both ≥ 0 and ≤ 0 therefor it will be zero
Am I thinking of the way forward correctly?
Any help towards a solution would be great!
Thanks