# Tagged Questions

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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### Bounded linear extension of an operator whose co-domain is incomplete.

Let $X$ be a normed space and $Y$ be a Banach space. The bounded linear extension theorem states that a bounded linear operator $T : M \to Y$, where $M \subseteq X$ is dense, can be extended to a ...
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### Uniform Boundedness hypothesis

The hypothesis includes stating that the linear operators T, from X to Y, in the family of operators are bounded; how is the statement that follows in the hypothesis any different? " If for all x in ...
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### Is K(H) separable?

Let $H$ be an inifinite dimensional separable K-Hilbert space with $K$ could be $\mathbb{R}$ or $\mathbb{C}$. Are the compact operators $K(H)$ separable? It's well known that $\overline{F(H)}=K(H)$, ...
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### Why is the dual cone of $l^1$ is $l^\infty$?

I just noticed somewhere in Convex Optimization that the dual cone of $l^1$ is $l^\infty$! (A diamond in $\mathbb{R}^2$ for $l^1$ is a square in $\mathbb{R}^2$ for $l^\infty$.) In fact I cannot ...
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### Hilbert Spaces and Hamel Basis

Let $H$ be a Hilbert Space of infinite dimension, $S$ a not finite orthonormal basis and $B$ a Hamel basis to $H$. i) How to show that the cardinality of $B$ is greater than or equal to the ...
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### Non-commutaive Gelfand-Naimark theorem and dimension of Hilbert space

It is well known that using non-commutative Gelfand-Naimark theorem for finite dimensional $C^∗$-algebra we can obtain isometric representation on finite dimensional Hilbert space. My question is : ...
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### closedness of a graph of an unbounded operator between two Banach spaces

Let T be a linear operator from a subset $D(T)$ of a Banach space $X$ to another Banach space $Y$. T is closed if a sequence $x_n$ in $D(T)$ converges to $x \in X$ and the sequence $Ax_n$ converges to ...
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### $A$ is a Hermitian operator on an infinite dimensional Hilbert space and $\langle Ax|x\rangle=0$ for all $x$, prove $A=0$ without the spectral theorem

If $A$ is a Hermitian operator on an infinite dimensional Hilbert space such that $\langle Ax|x\rangle=0$ for all vectors $x$, can we prove $A=0$ without the spectral theorem? The proof seems ...
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### Is it true that the closed convex hull of a compact subset of the dual equipped with the w*-topology is compact?

Let $X$ be a Banach space. Consider the dual $X^*$ equipped with the weak*-topology. Is it true that the closed convex hull of a compact subset $K$ of the dual $X^*$ is compact? ps: I know that the ...
### If $A_j$ is an increasing family of Hermitian operators such that $A_j\nearrow A$ weakly, for $A=\mathrm{LUB}A_j$, then $A_j\rightarrow A$ strongly.
I am trying to prove the following proposition from Berberian's 'Notes on Spectral Theory': Proposition 1: If ($A_j$) is an increasingly directed family of Hermitian operators, and if the family ...