# Is Symmetric operator linear

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 $$(Ax\vert y)=(x\vert Ay)$$ $$(x,y\in X)$$

• What is a symmetric operator? – copper.hat Aug 5 '14 at 16:01
• $(A(x_1 + x_2) | y) = (x_1 + x_2, Ay) = (Ax_1, y) + (Ax_2,y)$. If this holds for every $y$, then... – Christopher A. Wong Aug 5 '14 at 16:11
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 $z$, $A(x+ty) = Ax + tAy$.