Consider the following SDE
$$df(t,T)=\alpha (t,T)dt+\sigma(t,T) \cdot dW_t$$ where $W$ is a d-dimensional Wiener process and $0\leq t \leq T<T^*$ and $T, T^*$ are fixed. Or in an integral form the solution looks like $$f(t,T)=f(0,T)+\int_0^t\alpha(u,T)du+\int_0^t\sigma(u,T)\cdot dW_u, \forall t\in[0,T]$$
and $f(0,.):[0,T^*]\to \mathbb{R}$ is a Borel-measurable function and $\alpha, \sigma$ are sufficiently regular so that the integrals are well defined.
This maybe a stupid question. Can this equation be considered as a infinite dimensional SDE? This is the HJM equation to price interest rate derivatives and I have seen in many books referring to this as an infinite dimensional SDE. Why ?
It depends where your unknown process f takes values. If it takes values in infinite dimensional spaces like for example $L^2([0,T]; \mathbb{R})$, we speak of stochastic equation in infinite dimension. The definition of Wiener process has to be adapted consequently. It looks like your initial condition is indeed a function so the process $(f_t)_t$ takes values in the space of Borel measurable functions.
Classic reference for infinite dimensional sdes is the book by Da Prato and Zabczyk https://www.cambridge.org/core/books/stochastic-equations-in-infinite-dimensions/6218FF6506BE364F82E3CF534FAC2FC5