For normed spaces \(X\) and \(Y\) and a linear map \(T\colon X\to Y\), the operator norm is \[ \norm{T}\coloneqq\sup_{\norm{x}\le 1}\norm{Tx}. \] \(T\) is continuous if and only if it is continuous at \(0\), if and only if \(\norm T<\infty\). The dual space \(X^*\) of bounded linear functionals, with the operator norm, is a Banach space for every normed space \(X\), since \(\RR\) and \(\CC\) are complete.
Linear functionals are continuous if and only if bounded (Tag T-5IWCG)
Let \((X,\norm{\cdot})\) be a normed vector space over \(\CC\) and let \(L\colon X \to \CC\) be a linear functional. The following are equivalent:
\(L\) is continuous.
\(L\) is continuous at \(0\).
\(L\) is bounded: there exists \(c\geq 0\) such that \(\abs{L(x)} \leq c \norm{x}\) for all \(x\in X\).
The operator norm is a norm on the dual of a Hilbert space (Tag T-QBTWN)
Let \(H\neq\theset{0}\) be a Hilbert space over \(\CC\), and let \(\dualof{H}\) be its dual. Then the operator norm \(\norm{L}_{\mathrm{op}}\coloneqq\sup_{\norm{x}=1}\abs{L(x)}\) is a norm on the vector space \(\dualof{H}\).
The dual space $\dualof{X}$ with the operator norm is a Banach space (Tag T-W5SDY)
Let \(X\) be a normed vector space over \(\mathbb K\in\theset{\RR,\CC}\), and let \(\dualof{X}\) be the vector space of continuous linear functionals \(X\to\mathbb K\) with the operator norm \(\norm{L}_{\text{op}}\coloneqq\sup_{x\in X,\ \norm{x}\leq1}\abs{L(x)}\). Then \((\dualof{X},\norm{\cdot}_{\text{op}})\) is a Banach space.
Compact operators
Compact operators (Tag FF-BSYDE)
Let \(X\) and \(Y\) be normed vector spaces. A linear map \(T\colon X\to Y\) is compact if \(T(B)\) has compact closure in \(Y\) for every bounded subset \(B\subseteq X\).
Every compact linear map \(T\colon X\to Y\) between normed vector spaces is bounded.
The closure of \(T(\theset{x\in X\suchthat\norm{x}\leq 1})\) is compact, hence bounded, so \(\sup_{\norm{x}\leq 1}\norm{Tx}<\infty\).
Let \(H\) be a Hilbert space. The set of compact operators \(H\to H\) is the closure, in the operator norm, of the set of bounded operators \(H\to H\) of finite rank.
Every compact operator is bounded, since a relatively compact set is bounded. The identity map of an infinite-dimensional Banach space is bounded and not compact, since its closed unit ball is not compact.