In this entry, whenever $V,W$ are normed vector spaces, $\mathcal{B}(V,W)$ denotes the algebra of bounded linear operators $V \longrightarrow W$ .

Let $X, Y, Z$ be normed vector spaces and $X', Y', Z'$ be their continuous dual spaces. Let $T, S \in \mathcal{B}(X,Y)$ , $R \in \mathcal{B}(Y,Z)$ and $\lambda \in \mathbb{C}$ .

Basic properties

• $T' \in \mathcal{B}(Y',X')$ and $\|T\|=\|T\,'\|$ .
• $(\lambda T)' = \lambda T'$ .
• $(S+T)' = S'+T'$ .
• $(RT)' = T'R'$ .
• If $T^{-1}$ exists and $T^{-1} \in \mathcal{B}(Y,X)$ then $(T')^{-1} \in \mathcal{B}(X',Y')$ and $(T')^{-1} = (T^{-1})'$ .

Miscellaneous properties

• If we endow $X'$ and $Y'$ with the weak-* topology then $T':Y' \longrightarrow X'$ is continuous.
• $T$ is an isometric isomorphism if and only if $T'$ is an isometric isomorphism.
• If $T$ is compact then $T'$ is also compact.
• If $T'$ is compact and $Y$ is a Banach space, then $T$ is also compact.