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Math 6210 • Real Analysis

Summer 2017 • Question 5

Question

Math 6210 • Real Analysis Summer 2017 • Question 5

Prove the Banach-Steinhaus theorem. That is, show that if \((B,\lVert \cdot \rVert _B)\) and \((\hat B,\lVert \cdot \rVert _{\hat B})\) are Banach spaces and \(A_i:B\to \hat B\) are a sequence of continuous linear maps from \(B\) to \(\hat B\), then either there exists \(M\) so that \(\lVert A_i\rVert _{\mathrm {op}}<M\) for all \(i\), or there exists \(x\in B\) so that \[ \sup _i\lVert A_i x\rVert _{\hat B}=\infty . \]

Raw LaTeX

Prove the Banach-Steinhaus theorem. That is, show that if
\((B,\lVert\cdot\rVert_B)\) and \((\hat B,\lVert\cdot\rVert_{\hat B})\) are
Banach spaces and \(A_i:B\to\hat B\) are a sequence of continuous linear maps
from \(B\) to \(\hat B\), then either there exists \(M\) so that
\(\lVert A_i\rVert_{\mathrm{op}}<M\) for all \(i\), or there exists \(x\in B\) so
that
\[
\sup_i\lVert A_i x\rVert_{\hat B}=\infty.
\]

Topic Tags

Banach Spaces Lp Spaces and Sequence Spaces