Let $X$ and $Y$ be vector spaces over a field $\mathbb{F}$ and let $f:X\to Y$ be a linear map. If the dual map $f^*: Y^*\to X^∗:y^*\mapsto y^*\circ f$ is the zero map, is the original map $f$ the zero map too?
Does that fact that the dual map is zero imply that the map is zero?
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linear-algebra
functional-analysis