# Questions tagged [topological-vector-spaces]

A topological vector space is a vector space $V$ over a topological field $\mathbb{K}$ (typically $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), together with a topology on $V$ such that vector addition and scalar multiplication are both continuous. Hilbert spaces and Banach spaces are examples of topological vector spaces.

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### Fundamental biorthogonal systems in barreled spaces

Let $X$ be a barreled vector space (over the field of real or complex numbers), $X'$ its topological dual space, and $\langle\cdot,\cdot\rangle$ the canonical bilinear form on $X \times X'$. ...
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Let $D\subset X$ be a dense subset of a complete separable locally convex space $X$ over $\mathbb{R}$. Though the question seems simple enough, I can't seem to find the answer in the literature: If $... 8 votes 0 answers 178 views ### On "linearly independent" metric spaces Urysohn's universal metric space$\Bbb U$satisfies the following surprising property: Whenever$i\colon\Bbb U\to E$is an isometric embedding into a normed vector space such that$0\not\in i(\Bbb U)$... 1 vote 2 answers 273 views ### Topologies on space of compactly supported continuous functions Let$X$be a locally compact Hausdorff space. As far as I understand, the space$C_c(X) = C_c(X; \mathbb{C})$of compactly supported continuous complex-valued functions on$X$is (most?) often ... 6 votes 1 answer 242 views ### Is the tensor product of distributions a continuous bilinear map with respect to the weak topology? Let$X$and$Y$be smooth manifolds. The map$\mathcal{D}'(X)\times\mathcal{D}'(Y)\to\mathcal{D}'(X\times Y)$given by$(S,T)\mapsto S\boxtimes T$is continuous with respect to the strong topology. Is ... 2 votes 1 answer 155 views ### Biorthogonal weakly null basic sequences Let$E$be a Banach space, let$e_{n}\in E$and$g_{n}\in E^{*}$be biorthogonal basic sequences (i.e.$\left<e_n,g_m\right>=\delta_{mn}$). Moreover, both of these sequences are weakly null. (... 5 votes 0 answers 117 views ### Logarithm on formal power series continuous? Denote$V:=\mathbb{R}^d$and consider the Cartesian product$V^\infty:=\prod_{k=0}^\infty V^{\otimes k}$together with its canonical projections$\pi_k : V^\infty\rightarrow V^{\otimes k}, (v_0, v_1, \...
A Künneth formula by Grothendieck/Schwartz states the following: Let $A, B$ be chain complexes of nuclear Fréchet spaces. If the differentials $d_A, d_B$ are topological homomorphisms (meaning in ...