Intereting Posts

Prove every odd integer is the difference of two squares
There is no isometry between a sphere and a plane.
If $d=\gcd\,(f(0),f(1),f(2),\cdots,f(n))$ then $d|f(x)$ for all $x \in \mathbb{Z}$
pigeonhole principle divisibility proof
Would like a hint for proving $(\forall x P(x)) \to A \Rightarrow \exists x( P(x) \to A)$ in graphical proof exercise on The Incredible Proof Machine
the Stone-Čech compactification by using ultrafilters
Is there a systematic way of solving cubic equations?
Show that the lexicographic order topology for $\mathbb{N}\times \mathbb{N}$ is not the discrete
Questions about Combinatorial Proof of $\sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$
What is the probability of this exact same Champions League draw?
Is norm $E^{1/p}$ a continous function of $p$
How to calculate volume of 3d convex hull?
Conditional and Total Variance
How do I integrate $\int_{0}^{1}\!\sin x^2\,dx$?
Geometric distinction between real cubics with different Galois group?

Let $X$ be a Banach Space and $Y$ be a normed linear space. Show that if $T$ is an isometry then $T(X)$ is closed in $Y$.

Let me have some idea to solve this. Thank you for your help.

- Was Grothendieck familiar with Stone's work on Boolean algebras?
- References for Banach Space Theory
- Proving that Tensor Product is Associative
- An approximate eigenvalue for $ T \in B(X) $.
- How to proof that a finite-dimensional linear subspace is a closed set
- Compact subspace of a Banach space .

- Frechet differentiable implies reflexive?
- An approximate eigenvalue for $ T \in B(X) $.
- Is there a concept of a “free Hilbert space on a set”?
- Lower Semicontinuity Concepts
- Does the sequence $(\sqrt{n} \cdot 1_{})_n$ converge weakly in $L^2$?
- Weak limit of an $L^1$ sequence
- How to show this sequence is a delta sequence?
- Question about weak convergence, $\lbrace f(x_{n}) \rbrace$ converges for all $n$, then $x_{n} \rightharpoonup x$
- Do there exist two singular measures whose convolution is absolutely continuous?
- Any two norms equivalent on a finite dimensional norm linear space.

I had problem in showing every sequence is cauchy in T(X).Now it is done.

Let $y_{n}$ be sequence in $T(X)$. Since $y_{n}=T(x_{n})$ $\forall n$. So we have got a sequence $(x_{n})_{{n \in \mathbf{N}}}$ in X.

Now X is a Banach Space therefore it is complete and hence $x_{n} \longrightarrow x$ for some $x$ in $X$.

$\|T(x_{n})-T(x_{m})\|=\|T(x_{n}-x_{m})\|=\|x_{n}-x_{m}\|$ as $T$ is isometry.

Since $(x_{n})_{n \in \mathbf{N}}$ is convergent it is Cauchy. This tells that $(T(x_{n}))_{n \in \mathbf{N}}$ is cauchy. But T is continious so $T(x_{n}) \longrightarrow T(x)$ and uniqueness of limit point gives $y_{n} \longrightarrow T(x)$. It shows that every sequence in $T(X)$ converges in $T(x)$. Therefore T(X) is closed in Y.

- Question about the cardinality of a space
- A determinant inequality
- What is the tens digit of $3^{100}$?
- Find the total number of zeros in given range of decimal numbers
- Curve in $\mathbb{A}^3$ that cannot be defined by 2 equations
- Convert from Nested Square Roots to Sum of Square Roots
- Ideals of $\mathbb{Z}$
- Calculation of Bessel Functions
- $x^p -x-c$ is irreducible over a field of characteristic $p$ if it has no root in the field
- Embed $S^{p} \times S^q$ in $S^d$?
- If $\sum\limits_{k=1}^{\infty}a_k=S $, then $ a_4+a_3+a_2+a_1+a_8+a_7+a_6+a_5+\dots=?$
- Problem with my floor…
- group presentation and the inverse elements of the generators
- Sum of odd Bessel Functions
- largest fraction less than 1