Adjoint operator definition 
Is an adjoint operator only applicable in Hilbert space?

Every definition on the internet is different from my teacher's definition.
My teacher's definition:
$T$ being an element of the continuous linear application from $X$ to $Y$, the adjoint operator of $T$ is $T^*$, element of the continuous linear application between $Y^*$ and $X^*$:
$$T^*:Y^* \,\text{ to }\, X^*\;\; (g \,\text{ to }\, T^*g).$$
So nowhere it talks about Hilbert space. Is the definition lacking something?

In Brezis's Functional Analysis the adjoint of a linear operator is defined the following way:
Let $X$, $Y$ be normed spaces and let $L:X \longrightarrow Y$ be a linear bounded operator. The adjoint operator $L^{*}: Y^{*} \longrightarrow X^{*}$ is defined by:
$\langle L^{*},y^{*}\rangle = \langle y^{*}, L \hspace{0.1cm} . \rangle $ 
With $X^{*}$, $Y^{*}$ the dual spaces of $X$ and $Y$ respectively. It is important to note that $\langle y^{*}, L \hspace{0.1cm} . \rangle $ is a function in $X^{*}$, i.e. $\langle y^{*}, L \hspace{0.1cm} . \rangle : X \longrightarrow \mathbb{R} $.
The notation $ \langle f,x \rangle $ is used for $f(x)$.
