# Infimum or supremum of inner products

I sometimes see following objects in papers, $$\sup_{r\in K} \langle x,r\rangle$$ or $$\inf_{r\in K} \langle x,r\rangle$$

I know its meaning changes according to the set $K$, but I would like to gain insight when I see these type of things. What they mean actually?

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Remember the inner product's relationship to angles between vectors in $\mathbb{R}^n$ $$cos(\theta)= \frac{\langle x, y \rangle}{\|x\|\|y\|}.$$
Assuming that the set $K$ is norm bounded what you are looking for is the vector in $K$ that points most in the direction of $x$ or most in the opposite direction (the direction of $-x$). In a general inner product space or Hilbert space the same intuition applies.