Weighted inner product space and representation of dual space

Question

Let $H$ be a Hilbert space and define $H_c$ to be the weighted Hilbert space with inner product $$(u,v)_{H_c} = c(u,v)_H$$ where $c$ is a positive constant.

Then is it true that $$c\langle f, u \rangle_{H^*, H} = \langle f, u\rangle_{H_c^*, H_c}$$? So basically I am asking if the Riesz maps are the same.