the modulus of smoothness of Hilbert space Let $(X, ||.||)$ be a Banach space. For $t&gt;0$, the modulus of
smoothness of $||.||$ is defined by
$$\rho_X(t)=\sup\left\{\frac{||x+ty||+||x-ty||}{2}-1: x,y\in
S_X\right\}.$$ Ican easily calculate
$$\rho_{H}(t)\leq\sqrt{1+t^2}-1.$$ Bu I can not prove $$\rho_{H}(t)\geq\sqrt{1+t^2}-1.$$ Please help me

thanks
$\def\norm#1{\left\|#1\right\|}\def\&lt;#1&gt;{\left(#1\right)}$
Let $H$ be a Hilbert space (if $H$ is a real Hilbert space, we need $\dim H \ge 2$), $x,y\in H$ and $t \ge 0$. We have for $x,y \in S_H$:
\begin{align*}
  \norm{x+ty} + \norm{x-ty} &amp;= \&lt;x+ty,x+ty&gt;^{1/2} + \&lt;x-ty,x-ty&gt;^{1/2}\\
     &amp;= \Bigl( \&lt;x,x&gt; + 2t\Re\&lt;x,y&gt; + t^2\&lt;y,y&gt;\Bigr)^{1/2} + \Bigl( \&lt;x,x&gt; - 2t\Re\&lt;x,y&gt; + t^2\&lt;y,y&gt;\Bigr)^{1/2}\\
     &amp;= \Bigl( 1+t^2 + 2t\Re\&lt;x,y&gt; \Bigr)^{1/2} + \Bigl( 1 + t^2 - 2t\Re\&lt;x,y&gt;\Bigr)^{1/2}\\
\end{align*}
Now the supremum of these terms is greater or equal to each of them, espacially to those where $\Re\&lt;x,y&gt; = 0$. Hence
$$ \sup_{\norm x = \norm y = 1} \norm{x+ty}+\norm{x-ty} \ge 2(1+t^2)^{1/2}. $$
Let H be a hilbert space then for  $t&gt;0$ 
\roh_H(t)=\sup\left{\frac{t ebsilon{2}-1+(1-ebsilon^2\frac{4})^{1/2}:ebsilon\ge 0 and less of 2}=\leq\sqrt{1+t^2}-1.and know \rho_X(t)=\sup\left{\frac{t\ebsilon\frac\2-\delta_X(ebsilon) :ebsilon between 0 and 2} and delta_X(ebsilon)={1-ebsilon^2\frac{4})^{1/2}}