Let $f:[a,b]\to H$ for an arbitrary real-valued Hilbert space $H$ be continuous on the real interval $[a,b]$. We are given that $B=\{b_k:k\in \mathbb{N}\}$ is an orthonormal basis for $H$. I am trying to show that $$\sum_{k=1}^{n}\langle f(t),b_k\rangle^2$$ converges uniformly to $\|f(t)\|^2$ on the interval.
I'm easily able to show pointwise convergence just using the fact that there is a sequence of scalars $(\alpha_k)_t$ such that for any $t$ in the interval $$f(t)=\sum_{k=1}^\infty\alpha_kb_k.$$ I know I need to bring in the continuity of $f$ to gain uniform convergence but I'm not sure how to do that in my present argument. Any tips would be appreciated.
Let's denote $$ f_n(t) := \sum_{k=1}^n \langle f(t), b_k \rangle^2. $$ Then, the $f_n$ are continuous, increasing and converge pointwise to the continuous function $g(t) := \lVert f(t) \rVert^2$. The fact that the $f_n$ are increasing implies uniform convergence.
See for example Sequence of monotone functions converging to a continuous limit, is the convergence uniform?.