Today reading a book functional analysis of this sentence stroke. please could you help someone to prove this means i.e. that: symmetric operator is linear, thank you preliminarily. Thanky very much

Definiton of the symmetric operator: Let be $X$ a unitary space. Operator $A:X\rightarrow X$ is symmetric if $$(Ax\vert y)=(x\vert Ay)$$ $$(x,y\in X)$$

  • $\begingroup$ What is a symmetric operator? $\endgroup$ – copper.hat Aug 5 '14 at 16:01
  • $\begingroup$ $(A(x_1 + x_2) | y) = (x_1 + x_2, Ay) = (Ax_1, y) + (Ax_2,y)$. If this holds for every $y$, then... $\endgroup$ – Christopher A. Wong Aug 5 '14 at 16:11
  • $\begingroup$ please help me, to prove this fact $\endgroup$ – user145717 Aug 5 '14 at 16:12
  • $\begingroup$ Consider what it means to be linear, and look at what I've written as a hint. $\endgroup$ – Christopher A. Wong Aug 5 '14 at 16:16
  • $\begingroup$ please help me because I did not understand very well these definitions, so I applied this site, believing that someone will help me $\endgroup$ – user145717 Aug 5 '14 at 16:18

I'll assume the "mathematician's" convention that the inner product is linear in the first variable and conjugate-linear in the second. For vectors $x$, $y$, $z$ and scalar $t$,

$$\eqalign{\langle A(x+ty) | z \rangle &= \langle (x+ty) | A z \rangle\cr & = \langle x | A z \rangle + t \langle y | A z \rangle\cr & = \langle Ax | z \rangle + t \langle A y | z \rangle\cr &= \langle Ax + tAy | z \rangle }$$

Since this is true for all $z$, $A(x+ty) = Ax + tAy$.


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