Let $X$ be a finite dimensional Banach space. A function $f:X\rightarrow \mathbb{R}$ is called convex, if $f\left( \left( 1-\lambda \right) x+ \lambda y\right) \leq \left( 1-\lambda \right) f\left( x\right) +\lambda f\left( y\right)$ for all $x,y\in X$ and $\lambda \in \left[ 0,1\right]$.
How can i show that a twice differentiable function $f:X\rightarrow \mathbb{R}$ is convex, iff $D^{2}f\left( x\right) \left( v,v\right) \geq 0$ for all $x,v\in X$.
I know how to prove this for $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ but I can‘t find a way to use this.