Duality pairing in Banach spaces
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Suppose we have a Banach space $X$ and its dual $X^{'}$ . Then the duality pairing is often written $$langle f,vrangle_{X^{'},X} = f(v)$$ for $f in X^{'}$ and $v in X$ . But is it allowed to write $$ langle v,frangle_{X,X^{'}} $$ to mean $langle f,vrangle_{X^{'},X}$ ? In the setting of a Hilbert space $H$ , for a given linear operator $A:Hto H^{'}$ , I saw $A$ being self-adjont is written as $$ langle Av,urangle = langle v,Aurangle $$ Does $langle v,Aurangle$ here actually mean $langle Au,vrangle= Au(v)$ ?
functional-analysis
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edited Jan 17 at 18:54
