Intereting Posts

Infinite square-rooting
Evaluation of $\sum^{\infty}_{n=0}\frac{1}{16^n}\binom{2n}{n}.$
$f: \Omega \rightarrow \Omega$ holomorphic, $f(0) = 0$, $f'(0) = 1$ implies $f(z) = z$
If every five point subset of a metric space can be isometrically embedded in the plane then is it possible for the metric space also?
Why do we say the harmonic series is divergent?
Asymptotic behavior of Harmonic-like series $\sum_{n=1}^{k} \psi(n) \psi'(n) – (\ln k)^2/2$ as $k \to \infty$
Homogeneous differential equation $\frac{dy}{dx} = \frac{y}{x}$ solution?
Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$
Infinite coproduct of rings
Why 6 races are not sufficient in the 25 horses, 5 tracks problem
Can all continuous linear operators on a function space be represented using integrals?
do eigenvectors correspond to direction of maximum scaling?
Classsifying 1- and 2- dimensional Algebras, up to Isomorphism
Primary ideals confusion with definition
What is a projective ideal?

I have a question.

If $X$ and $Y$ are Banach spaces, we have to prove that a compact linear operator is completely continuous.

A mapping $T \colon X \to Y$ is called completely continuous, if it maps a weakly convergent sequence in $X$ to a strongly convergent sequence in

$Y$ , i.e., $x_n\underset{n\to +\infty}\rightharpoonup x$ implies $\lVert Tx_n-

Tx\rVert_Y\to 0$.

- Bochner Integral vs. Riemann Integral
- Proving that Tensor Product is Associative
- Square root of compact operator
- Bochner: Lebesgue Obsolete?
- Non-closed subspace of a Banach space
- A few questions about the Hilbert triple/Gelfand triple
- Nested sequences of balls in a Banach space
- Banach space in functional analysis
- Absolutely convergent sums in Banach spaces
- Prove a certain property of linear functionals, using the Hahn-Banach-Separation theorems

Since it’s a homework question I will just give some steps.

- By linearity, we can assume that $x=0$.
- We have to show that for each subsequence of $\{Tx_n\}$, we can extract a further subsequence which converges to $0$ in norm in $Y$.
- A weakly converging sequence is bounded.
- $T$ maps bounded sets to sets with a compact closure.

Once the second steps is shown, we can conclude. Indeed, assume that $Tx_n$ doesn’t converge to $0$. Then we are able to find $\delta>0$ and $A$ an infinite subset of the natural numbers such that $\lVert Tx_k\rVert_Y\geq\delta$ for each element of $A$. We can consider it as a subsequence, and we can’t extract a further subsequence which converges to $0$, a contradiction.

- What does the symbol $\lll$ mean?
- Intuition for Smooth Manifolds
- Closed-form of $\int_0^1 B_n(x)\psi(x+1)\,dx$
- Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)
- What is the chance that a PDF with compact support is concave?
- Alternative set theories
- How prove this sequence $S_{n}=,n\in N$ contains infinitely many composite numbers
- Why are projective spaces and varieties preferable?
- Krull dimension of this local ring
- Examples of transfinite induction
- Total Variation and indefinite integrals
- Convergence of Ratio Test implies Convergence of the Root Test
- Metric Space and ordered field
- Are Cumulative Distribution Functions measurable?
- Sum of closed and compact set in a TVS