Let $ X$ be a Banach space and $ M$ be a closed subspace i.e. $ M \leq X$ .

Is it true that $ M^*$ is a subspace of $ X^*$ where $ *$ denotes the dual space?

What I have tried:

I know by Hahn Banach theorem that every $ f \in M^*$ has an extension $ \hat f \in X^*$ such that $ \hat f |_M =f$ and the operator norm doesn’t change i.e. $ \|\hat f\| =\|f\|$ .

I found that such an extension can be chosen to be linear if $ X$ is a Hilbert space here, hence my claim might not hold in the general case.

Any help is appreciated.