Adjoint operator

$\textbf{Adjoint operator}$ Let $X,Y$ be normed spaces and $T\in \mathcal{B}(X,Y)$ a bounded linear operator from $X$ to $Y$. Then the adjoint operator $T^*:Y^*\to X^*$ is defined by $$(T^*y^*)(x)=y^*(Tx) \qquad y^*\in Y^*, x \in X.$$ Note: $T^*\in\mathcal{B}(Y^*,X^*)$, i.e. $T^*$ is a bounded linear operator.
$\textbf{Adjoint operator}$ Let $X,Y$ be normed spaces and $T\in \mathcal{B}(X,Y)$ a bounded linear operator from $X$ to $Y$. Then the adjoint operator $T^*:Y^*\to X^*$ is defined by $$(T^*y^*)(x)=y^*(Tx) \qquad y^*\in Y^*, x \in X.$$ Note: $T^*\in\mathcal{B}(Y^*,X^*)$, i.e. $T^*$ is a bounded linear operator.
copied
𝐀𝐝𝐣𝐨𝐢𝐧𝐭 𝐨𝐩𝐞𝐫𝐚𝐭𝐨𝐫 Let 𝑋,𝑌 be normed spaces and 𝑇∈ℬ(𝑋,𝑌) a bounded linear operator from 𝑋 to 𝑌. Then the adjoint operator 𝑇*:𝑌*→𝑋* is defined by (𝑇*𝑦*)(𝑥)=𝑦*(𝑇𝑥) 𝑦*∈𝑌*,𝑥∈𝑋. Note: 𝑇*∈ℬ(𝑌*,𝑋*), i.e. 𝑇* is a bounded linear operator.
copied