[proofplan]
The forward implication is an application of the [continuous mapping theorem](/theorems/1847) to each continuous linear functional $x\mapsto a\cdot x$. For the converse, the assumed convergence of all scalar projections gives convergence of the one-dimensional characteristic functions of $u\cdot Y_n$ for every direction $u\in\mathbb R^k$. These one-dimensional characteristic functions are exactly the multivariate characteristic functions of $Y_n$ evaluated at $u$, so Lévy's [continuity theorem](/theorems/1145) in $\mathbb R^k$ yields convergence in distribution of the random vectors.
[/proofplan]
[step:Apply the continuous mapping theorem to each linear projection]
Assume first that $Y_n\xrightarrow{d}Y$ in $\mathbb R^k$. Fix $a\in\mathbb R^k$, and define the linear projection
\begin{align*}
\pi_a:\mathbb R^k\to\mathbb R,\qquad \pi_a(x)=a\cdot x.
\end{align*}
The map $\pi_a$ is continuous with respect to the Euclidean topologies. Hence, by the continuous mapping theorem (citing a result not yet in the wiki: Continuous Mapping Theorem),
\begin{align*}
\pi_a(Y_n)\xrightarrow{d}\pi_a(Y).
\end{align*}
Since $\pi_a(Y_n)=a\cdot Y_n$ and $\pi_a(Y)=a\cdot Y$, this proves
\begin{align*}
a\cdot Y_n\xrightarrow{d}a\cdot Y.
\end{align*}
Because $a\in\mathbb R^k$ was arbitrary, the forward implication follows.
[/step]
[step:Convert convergence of projections into convergence of characteristic functions]
Assume conversely that, for every $a\in\mathbb R^k$,
\begin{align*}
a\cdot Y_n\xrightarrow{d}a\cdot Y.
\end{align*}
For each $n\in\mathbb N$, define the characteristic function of $Y_n$ as the map
\begin{align*}
\varphi_n:\mathbb R^k\to\mathbb C,\qquad \varphi_n(u)=\mathbb E[\exp(iu\cdot Y_n)].
\end{align*}
Define the characteristic function of $Y$ as the map
\begin{align*}
\varphi:\mathbb R^k\to\mathbb C,\qquad \varphi(u)=\mathbb E[\exp(iu\cdot Y)].
\end{align*}
Fix $u\in\mathbb R^k$. Applying the hypothesis with $a=u$ gives
\begin{align*}
u\cdot Y_n\xrightarrow{d}u\cdot Y.
\end{align*}
The one-dimensional characteristic function convergence theorem, equivalently the one-dimensional Lévy continuity theorem (citing a result not yet in the wiki: Lévy continuity theorem for characteristic functions), gives
\begin{align*}
\mathbb E[\exp(it(u\cdot Y_n))]\to \mathbb E[\exp(it(u\cdot Y))]
\end{align*}
for every $t\in\mathbb R$. Taking $t=1$ yields
\begin{align*}
\varphi_n(u)\to\varphi(u).
\end{align*}
Thus $\varphi_n(u)\to\varphi(u)$ for every $u\in\mathbb R^k$.
[guided]
We now prove that the scalar projection hypothesis forces convergence of the multivariate characteristic functions. For each $n\in\mathbb N$, the characteristic function of the random vector $Y_n$ is the map
\begin{align*}
\varphi_n:\mathbb R^k\to\mathbb C,\qquad \varphi_n(u)=\mathbb E[\exp(iu\cdot Y_n)].
\end{align*}
Similarly, the characteristic function of $Y$ is
\begin{align*}
\varphi:\mathbb R^k\to\mathbb C,\qquad \varphi(u)=\mathbb E[\exp(iu\cdot Y)].
\end{align*}
Fix a direction $u\in\mathbb R^k$. The assumption of the theorem applies to every vector $a\in\mathbb R^k$, so it applies in particular to $a=u$. Therefore the real-valued random variables $u\cdot Y_n$ converge in distribution to the real-valued [random variable](/page/Random%20Variable) $u\cdot Y$:
\begin{align*}
u\cdot Y_n\xrightarrow{d}u\cdot Y.
\end{align*}
The one-dimensional characteristic function convergence theorem says that convergence in distribution of real-valued random variables implies pointwise convergence of their characteristic functions. Applying that theorem to the real-valued random variables $u\cdot Y_n$ and $u\cdot Y$, we obtain, for every $t\in\mathbb R$,
\begin{align*}
\mathbb E[\exp(it(u\cdot Y_n))]\to \mathbb E[\exp(it(u\cdot Y))].
\end{align*}
We only need the value $t=1$. Substituting $t=1$ gives
\begin{align*}
\mathbb E[\exp(iu\cdot Y_n)]\to \mathbb E[\exp(iu\cdot Y)].
\end{align*}
By the definitions of $\varphi_n$ and $\varphi$, this is exactly
\begin{align*}
\varphi_n(u)\to\varphi(u).
\end{align*}
Since the direction $u\in\mathbb R^k$ was arbitrary, the characteristic functions $\varphi_n$ converge pointwise to $\varphi$ on all of $\mathbb R^k$.
[/guided]
[/step]
[step:Apply Lévy continuity theorem in $\mathbb R^k$]
The function $\varphi$ is the characteristic function of the $\mathbb R^k$-valued random vector $Y$, hence it is continuous at $0\in\mathbb R^k$. From the previous step,
\begin{align*}
\varphi_n(u)\to\varphi(u)
\end{align*}
for every $u\in\mathbb R^k$. By Lévy's continuity theorem in $\mathbb R^k$ (citing a result not yet in the wiki: multivariate Lévy continuity theorem for characteristic functions), pointwise convergence of the characteristic functions $\varphi_n$ to the characteristic function $\varphi$ implies
\begin{align*}
Y_n\xrightarrow{d}Y.
\end{align*}
This proves the converse implication and completes the proof.
[/step]
