In the first half of the article, we introduce the notion of the universal unitary completion of a continuous representation of a $p$ -adic reductive group on a locally convex $p$ -adic vector space, and we prove that such a completion exists under appropriate hypotheses. The problem of studying unitary completions has been raised by Breuil [4] in connection with his work on a possible $p$ -adic local Langlands correspondence for $GL_2$ , and we relate our construction to certain conjectures of Breuil in [4, Sec. 1.3] for the group $GL_2(\Q_p)$ . In particular, we show that the universal unitary completion of the locally analytic parabolic induction of a locally algebraic character coincides with the universal unitary completion of the corresponding locally algebraic induction, provided that the character being induced satisfies a noncritical slope condition (see Prop. 2.5). In the second half of the article, we consider a certain unitary Banach space representation of $GL_2(\Q_p)$ obtained by $p$ -adically completing the cohomology of classical modular curves. The mere existence of this representation implies that those locally algebraic, parabolically induced representations of $GL_2(\Q_p)$ which arise from classical finite-slope newforms have a nontrivial universal unitary completion (verifying a conjecture of Breuil in [4, Sec. 1.3] for these representations), while applying Proposition 2.5 in this context enables us to give a new construction of $p$ -adic $L$ -functions attached to $p$ -stabilized newforms of noncritical slope. Combining our construction with the representation-theoretic definition of ${\cal L}$ -invariants implicit in [5, Cor. 1.1.7], we are able to give a simple proof of the Mazur-Tate-Teitelbaum exceptional zero conjecture in [18, p. 46] (in terms of Breuil's definition of the ${\cal L}$ -invariant)