Equivalence of Definitions of Norm of Linear Functional
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Contents |
Theorem
Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$.
Then the following definitions of the norm of $L$ are equivalent:
- $(1): \qquad \left\|{L}\right\| = \sup \left\{{\left|{Lh}\right|: \left\|{h}\right\| \le 1}\right\}$
- $(2): \qquad \left\|{L}\right\| = \sup \left\{{\left|{Lh}\right|: \left\|{h}\right\| = 1}\right\}$
- $(3): \qquad \left\|{L}\right\| = \displaystyle \sup \left\{{\dfrac {\left|{Lh}\right|} {\left\|{h}\right\|}: h \in H, h \ne \mathbf 0}\right\}$
- $(4): \qquad \left\|{L}\right\| = \inf \left\{{c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|}\right\}$
Corollary
For all $h \in H$, the following inequality holds:
- $\left|{Lh}\right| \le \left\|{L}\right\| \left\|{h}\right\|$
Proof
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Sources
- John B. Conway: A Course in Functional Analysis (1990)... (previous)... (next) $I.3.3$
