linear algebra – Different definition for L smooth function.

At optimization class, professor gave the definition of L smooth function by

$$f:mathbf{R^n} rightarrow mathbf{R}$$ is L smooth if all the eigenvalue for $$nabla^{2} f$$ is smaller than L

where $$nabla^{2} f = begin{pmatrix} frac{partial ^2 f}{partial x_1 partial x_1} & frac{partial ^2 f}{partial x_1 partial x_2 } & cdots & frac{partial ^2 f}{partial x_1 partial x_n } \ vdots & vdots & ddots & vdots \ frac{partial ^2 f}{partial x_n partial x_1} & frac{partial ^2 f}{partial x_n partial x_2 } & cdots & frac{partial ^2 f}{partial x_n partial x_n } end{pmatrix}$$

but when I search for other books, it state that

$$f$$ is L smooth if $$frac{lVert nabla f(x)- nabla f(y)rVert}{lVert x-y rVert} leq L,$$ $$forall x,y in mathbf{R^n}$$

Are these two statement equivalent ?

What I have done is
$$frac{lVert nabla f(x)- nabla f(y)rVert}{lVert x-y rVert} leq L,$$ $$forall x,y in mathbf{R^n}$$ $$implies$$ $$lvert frac{partial^2 f}{partial x_i partial x_j} rvert leq L, forall i,j$$, but I don’t know how it relate to the eigenvalue of $$nabla^2f$$.

$$f(x^t)-f(x^*)=frac{1}{2}(x^t-x^*)^Tnabla^2f(bar{x})^T(x^t-x^*)leq frac{L}{2}lVert x^t-x^* rVert^2$$
where $$xin mathbf{R^n},$$ $$x^t$$ is t iteration for x, $$x^*$$ is the final form of $$x$$ s.t. $$f(x^*)$$ smallest, $$bar{x}$$ is some point between $$x^t$$ and $$x^*$$.
This part $$f(x^t)-f(x^*)=frac{1}{2}(x^t-x^*)^Tnabla^2f(bar{x})^T(x^t-x^*)$$ is by Tyler expansion which I can understand
$$frac{1}{2}(x^t-x^*)^Tnabla^2f(bar{x})^T(x^t-x^*)leq frac{L}{2}lVert x^t-x^* rVert^2$$ is where I don’t get it. Is it by the definition professor stated for L smooth ?