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 $nabla^2f$.

Later the professor had this

$f(x^t)-f(x^*)=frac{1}{2}(x^t-x^*)^Tnabla^2f(bar{x})^T(x^t-x^*)leq frac{L}{2}lVert x^t-x^* rVert^2$

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

The second inequation
$frac{1}{2}(x^t-x^*)^Tnabla^2f(bar{x})^T(x^t-x^*)leq frac{L}{2}lVert x^t-x^* rVert^2$ is where I don’t get it. Is it by the definition professor stated for L smooth ?