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.

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

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