Kazhdan's Property (T) and Subrepresentations I am trying to prove that $G$ having Kazhdan's Property (T) implies that whenever a unitary representation $(\pi, \mathcal{H})$ of $G$ weakly contains $1_G$, it contains $1_G$. 
Here is the definition of containment from the book I am using:

A unitary representation $\rho$ of $G$ is strongly contained or contained in a representation $\pi$ of $G$ if $\rho$ is equivalent to a subrepresentation of $\pi$.

What exactly is a subrepresentation of $\pi$? I searched the internet and all I could find were definitions about invariant subspaces of a representation. But the above definition seems to be using the word "subrepresentation" in a different sense. And being equivalent has something to do with the existence of a certain isometry between the Hilbert spaces, right? 
 A: B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), Cambridge University Press (2008), p. 290 (Section A.1):

Let $(\pi,\mathcal{H})$ be a unitary representation of a topological group $G$ in a Hilbert space $\mathcal{H}$, and let $\mathcal{K}$ be a closed $G$-invariant subspace of $\mathcal{H}$. Denoting, for every $g$ in $G$, by $\pi^\mathcal{K}(g)\colon\mathcal{K}\to\mathcal{K}$ the restriction of the operator $\pi(g)$ to $\mathcal{K}$, we obtain a unitary representation $\pi^\mathcal{K}$ of $G$ on $\mathcal{K}$. Then $\pi^\mathcal{K}$ is a subrepresentation of $\pi$.
An intertwining operator between two unitary representations $(\pi_1, \mathcal{H}_1)$ and $(\pi_2, \mathcal{H}_2)$ of $G$ is a continuous linear operator $T$ from $\mathcal{H}_1$ to $\mathcal{H}_2$ such that $T\pi_1(g) = \pi_2(g)T$ for all $g \in G$. The representations $\pi_1$ and $\pi_2$ are equivalent if there exists an intertwining operator $T \in \mathcal{L}(\mathcal{H}_1, \mathcal{H}_2)$ which is isometric and onto.

