Kerf

Let f be a linear functional of a normed space X = (E, ||·||). Suppose ker f is closed and let xn → x. Let yn be a convergent sequence lying entirely within ker f . Then yn → y where y ∈ ker f . Let v = x − y ⇒ x = v + y. Then xn can be written as xn = v + yn which gives us the following. x n = v + yn → v + y = x Now consider f (x) and f (xn ). f (x) = f (v + y) = f (v) f (xn ) = f (x + yn ) = f (v) Thus f (xn ) → f (x) and so f is continuous.

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