Let $\mathcal{H}$ denote a separable Hilbert space, with orthonormal basis $\{v_k\}_{k=1}^\infty$. Let $N \in \mathbb{N}$, and consider the subspace $M := \text{span} \{v_k\}_{k=1}^\infty$.
Show that for any $f \in \mathcal{H}$ and any coefficients $c_1,...,c_N \in \mathbb{C}$, \begin{equation*} \left\| f- \sum_{k=1}^N c_kv_k \right\|^2 = \sum_{k=1}^N |c_k - \langle f,v_k \rangle |^2 + \sum_{k=N+1}^\infty |\langle f,v_k \rangle |^2 \end{equation*}
My attempt so far \begin{align} \left\| f- \sum_{k=1}^N c_kv_k \right\|^2 & = \left\langle f- \sum_{k=1}^N c_kv_k, f- \sum_{k=1}^N c_kv_k \right\rangle \\ & = \left\langle f, f- \sum_{k=1}^N c_kv_k \right\rangle -\sum_{k=1}^N c_k \left\langle v_k, f- \sum_{k=1}^N c_kv_k \right\rangle \\ & = \langle f,f \rangle - \sum_{k=1}^N \overline{c}_k \langle f,v_k \rangle - \sum_{k=1}^N \overline{c}_k \langle v_k,f \rangle + \sum_{k=1}^N \sum_{l=1}^N \overline{c}_l c_k \langle v_k,v_l \rangle \end{align} I'm not really sure if I'm on the right track. Can somebody help me with the next step. Thanks.