Weak vs. strong convergence in the proof of the Hodge decomposition theorem in Warner, p.224

I'm reading the proof of the Hodge decomposition theorem in Warner, Foundations of Differentiable Manifolds and Lie Groups. At p.224, it is shown that $$\lim_{j\to\infty} \langle\beta_j,\psi\rangle = \langle\beta,\psi\rangle,\qquad\forall\psi\in E^p(M)$$ Then, Warner says :

Consequently, $$\beta_j \to \beta$$

without specifying if this is a strong or a weak convergence. It seems that it should be a weak convergence. But then, the sentence after is :

Since $$\|\beta_j\|=1$$ and $$\beta_j\in (H^p)^\perp$$, it follows that $$\|\beta\|=1$$ and $$\beta\in (H^p)^\perp$$.

which needs strong convergence. What am I missing?

A bit of context : we are on a smooth Riemannian oriented closed manifold. $$E^p(M)$$ denotes the space of differential $$p$$-forms on $$M$$. $$H^p$$ denotes the kernel of the Hodge-de Rham Laplacian $$\Delta=d\delta+\delta d$$. $$(H^p)^\perp$$ denotes the $$L^2$$-perpendicular to $$H^p$$. $$(\beta_j)$$ is a Cauchy sequence in $$(H^p)^\perp$$.

For the whole picture, the best is to read the page of the book.

• What are $H^p, E^p$? smooth functions defined around $p$ or... ? – user25959 Dec 3 '18 at 1:44
• @user25959 I added a bit of context in the question. – NAC Dec 3 '18 at 1:50