Existence of a state $\varphi$ over $\mathbb{B}(\mathcal{H})$ Let $T_1,T_2\in \mathbb{B}(\mathcal{H})$ be such that
$$\|T_1+\lambda T_2\|=\|T_1\|+\|T_2\|,$$
for some $\lambda \in \mathbb{C}$ with $|\lambda|=1$.
Iquote from a paper the following proof: It is well-known here is a state $\varphi$ over $\mathbb{B}(\mathcal{H})$ such that
$$\varphi((T_1+\lambda T_2)^*(T_1+\lambda T_2))=\|(T_1+\lambda T_2)^*(T_1+\lambda T_2)\|=\|T_1+\lambda T_2\|^2=(\|T_1\|+\|T_2\|)^2.$$
Thus
\begin{align*}
(\|T_1\|+\|T_2\|)^2&=\varphi(T_1^* T_1+\lambda T_1^* T_2+\overline{\lambda}T_2^* T_1+T_2^* T_2)
\\&=\varphi(T_1^* T_1)+\varphi(\lambda T_1^* T_2+\overline{\lambda}T_2^* T_1)+\varphi(T_2^* T_2)
\\&\leq\|T_1^* T_1\|+\|\lambda T_1^* T_2+\overline{\lambda}T_2^* T_1\|+\|T_2^* T_2\|
\\&=\|T_1^* T_1\|+\|T_1^* T_2\|+\|T_2^* T_1\|+\|T_2^* T_2\|
\\&\leq\|T_1\|^2+2\|T_1\|\,\|T_2\|+\|T_2\|^2
\\&=(\|T_1\|+\|T_2\|)^2.
\end{align*}

I don't understand the following two facts:

*

*What we mean by a state $\varphi$ over $\mathbb{B}(\mathcal{H})$?


*Why
$$\varphi(T_1^* T_1)+\varphi(\lambda T_1^* T_2+\overline{\lambda}T_2^* T_1)+\varphi(T_2^* T_2)
\leq\|T_1^* T_1\|+\|\lambda T_1^* T_2+\overline{\lambda}T_2^* T_1\|+\|T_2^* T_2\|\;?$$

 A: Suppose you have a $C^\ast$-algebra $A$. Then a state $\varphi \in A'$ is a functional such that $\lVert \varphi \rVert = 1$ and $\varphi(a^\ast a) \geq 0$ for all $a \in A$. 
One could also say that a state is a positive functional on $A$ with norm $1$ because $x \in A$ is positive if and only if there is $a \in A$ such that $x = a^\ast a$.
Finally, why do you find a state such that $\varphi(a)=\|a\|$ for each $a \in A$? 
This is basicly a consequence of the Hahn-Banach theorem. Clearly you find a functional of norm $1$ with that property by Hahn-Banach but you need to show that you can even find a state with that property. But this should be covered in any book about $C^*$-algebras. It works similarly as the construction of positive, norm attaining functionals on Banach lattices. 
Now $\mathbb B(\mathcal H)$ is a $C^\ast$-algebra where the involution is given by taking the adjoint of an operator. Hence a state on $\mathbb B(\mathcal H)$ is just a functional $\varphi: \mathbb B(\mathcal H) \to \mathbb C$ with $\lVert \varphi \rVert = 1$ and $\varphi(T^\ast T) \geq 0$ for all $T \in \mathbb B(\mathcal H)$. In particular, by Hahn-Banach for each $T \in \mathbb B (\mathcal H)$ there is a state $\varphi$ such that $\varphi(T)=\|T\|$ for each $T \in \mathbb B (\mathcal H)$.
The answer to your second question now follows from the fact that
$$ 
\varphi(T) \leq \lvert \varphi(T) \rvert \leq \lVert \varphi \rVert \lVert T \rVert = \lVert T \rVert \qquad (T \in \mathbb B(\mathcal H)).
$$
I hope that answers your question :) 
