Articles of functional analysis

$T$ is continuous if and only if $\ker T$ is closed

Let $X,Y$ be normed linear spaces. Let $T: X\to Y$ be linear. If $X$ is finite dimensional, show that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\ker T$ is closed. I am able to show that $X$, finite dimensional $\implies$ $T$ is bounded, hence continuous. […]

Boundedness and pointwise convergence imply weak convergence in $\ell^p$

Let $p\in(1,+\infty)$ and consider the space $\ell^p$ with its usual norm. The following are equivalent: (1) $x_n \rightharpoonup x$ (i.e. $x_n$ weakly converges to $x$); (2) $$\exists M>0: \quad \sup_{n} \Vert x_n \Vert_p \le M \quad \text{ and } \quad \forall k\in\mathbb N: \,\,\ x^{(k)}_n\to x^{(k)}$$ I think I’ve proved (1) $\Rightarrow$ (2): indeed, every […]

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by the Schr√∂dinger equation $$i\hbar\frac{\partial}{\partial t}|\psi_t\rangle = H|\psi_t\rangle$$ where $H$ is the Hamilton operator (for the free particle we have $H=-\frac{\hbar^2}{2m}\Delta$). Now I have often seen used spaces like […]

Compact operators on an infinite dimensional Banach space cannot be surjective

I am reading a book about functional analysis and have a question: Let $X$ be a infinite-dimensional Banach-space and $A:X \rightarrow X$ a compact operator. How can one show that $A$ can not be surjective?

Weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$

What does weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$ means? $\Omega$ is open, bounded, has boundary smooth and etc…

If for every $v\in V$ $\langle v,v\rangle_{1} = \langle v,v \rangle_{2}$ then $\langle\cdot,\cdot \rangle_{1} = \langle\cdot,\cdot \rangle_{2}$

Let $V$ be a vector space with a finite Dimension above $\mathbb{C}$ or $\mathbb{R}$. How does one prove that if $\langle\cdot,\cdot\rangle_{1}$ and $\langle \cdot, \cdot \rangle_{2}$ are two Inner products and for every $v\in V$ $\langle v,v\rangle_{1}$ = $\langle v,v\rangle_{2}$ so $\langle\cdot,\cdot \rangle_{1} = \langle\cdot,\cdot \rangle_{2}$ The idea is clear to me, I just can’t […]

Contraction mapping in an incomplete metric space

Let us consider a contraction mapping $f$ acting on metric space $(X,~\rho)$ ($f:X\to X$ and for any $x,y\in X:\rho(f(x),f(y))\leq k~\rho(x,y),~ 0 < k < 1$). If $X$ is complete, then there exists an unique fixed point. But is there an incomplete space for which this property holds as well? I think $X$ should be something […]

The direct sum of two closed subspace is closed? (Hilbert space)

I know that if $X$ is a Banach space, then, the direct sum of two closed subspace $X_1$ and $X_2$ is not necessarily closed. But what if $X$ is Hilbert? I assume there is something to do with the orthonormal basis, since this is something you can ask for a Hilbert space but not a […]

Isometric Embedding of a separable Banach Space into $\ell^{\infty}$

The problem is: Let $X$ be a separable Banach space then there is an isometric embedding from $X$ to $\ell^{\infty}$. My efforts: I showed that there is an isometry from $X^*$ (topological dual) to $\ell^\infty$ in the following way: Let $(e_{i})_{i=1}^{\infty}$ be a dense sequence in $B_{X}$ then define $\Phi:X^*\rightarrow\ell^\infty$ by $\Phi(f)=(f(e_{i}))_{i=1}^\infty$. It is clear […]

Is Banach-Alaoglu equivalent to AC?

The Banach-Alaoglu theorem is well-known. It states that the closed unit ball in the dual space of a normed space is $\text{wk}^*$-compact. The proof relies heavily on Tychonoff’s theorem. As I have recently figured out thanks to the nice guys on the chat belonging to this website is that Tychonoff’s theorem is equivalent to the […]