I am doing some work and reached an inequality where I think I should be able to go further but I am not sure how. I have a Hilbert space of functions on a set $X$. What I have: $$\forall x, y \in X, \langle f, g(x,\cdot)-g(y,\cdot) \rangle \leq || g(x,\cdot) -g(y,\cdot)|| $$ where the norm is induced by the inner product. Is there anything I can say about the norm of f?
2026-03-25 08:07:05.1774426025
Inner Product-Norm Inequality
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No, because $f$ can be orthogonal to $g$, then inner product between $f$ and $g$ doesn't tell you anything about the norm(length) of $f$