Here's an extract from some Continuous Martingales notes
I can see how K-W implies the blue box inequality but how does that inequality give continuity? Also what is the functional theorem that requires this continuity?
2026-04-04 10:59:34.1775300374
Proving existence of Itō Integral
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First, the statement that a bounded linear operator (or functional, in this case) on a normed space is continuous.
And second, the Riesz representation theorem. The easy direction of that theorem is the Cauchy–Schwarz inequality