Vector-valued function
Functions with values in a Hilbert space

If the argument of f is a real number and X is a Hilbert space, then the derivative of f at a point t can be defined as in the finite-dimensional case:

Most results of the finite-dimensional case also hold in the infinite-dimensional case too, mutatis mutandis. Differentiation can also be defined to functions of several variables (e.g., t ∈ R n {\displaystyle t\in \mathbb {R} ^{n}} or even t ∈ Y {\displaystyle t\in Y}, where Y is an infinite-dimensional vector space).

N.B. If X is a Hilbert space, then one can easily show that any derivative (and any other limit) can be computed componentwise: if

(i.e., f = f 1 e 1 + f 2 e 2 + f 3 e 3 + ⋯ {\displaystyle \mathbf {f} =f_{1}\mathbf {e} _{1}+f_{2}\mathbf {e} _{2}+f_{3}\mathbf {e} _{3}+\cdots }, where e 1, e 2, e 3, … {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3},\ldots } is an orthonormal basis of the space X), and f ′ (t) {\displaystyle f'(t)} exists, then

However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual topology of the Hilbert space.
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