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How can I show that if $\nu_{n,d}:\mathbb{P}_k^n\rightarrow \mathbb{P}_k^{\binom{n+d}{d}-1}$ denotes the Veronese map, then $\nu_{n,d}(\mathbb{P}_k^n)$ is a closed irreducible variety?

I have shown that if $X^j:=X_0^{j_0}\cdots X_N^{j_N}$, where $\sum_{l=1}^Nj_l=d$, then $$\nu_{n,d}(\mathbb{P}_k^n)=V\left(\{X^iX^j-X^kX^l:i+j=k+l\}\right),$$ which implies that it is a closed variety, but how can I prove that it is irreducible?

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    I guess you can show the image under a continuous map of an irreducible is irreducible. This is just a general topology fact. – 2017-02-25
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    Moreover, $\nu_{n,d}(\mathbb P^n_k)$ is actually isomorphic to $\mathbb P_k^n$. – 2017-02-25

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