# Tagged Questions

The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.

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### About a Weak Topology of a Vector Space

Let $X$ be a real vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each $p\in P$ ...
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### About a Weak Topology on TVS(part 2)

Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
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### If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?

Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive? To prove the contrapositive, it will suffice to assume that $X$ ...
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### Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
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### show $\langle f,g \rangle _w = \int^b_a f(x)g(x)dx$ is an inner product

Let $w(x)$ be a strictly positive continuous function on [a,b]. Define a form on $C[a,b]$ by the formula $\langle f,g \rangle _w = \int^b_a f(x)g(x)dx$ for $f,g \in C[a,b]$. Show that it is an inner ...
### Show that the norm of the derivitive of a $C^1$ function over a vector space is non-negative, homogeneous and satisfies the triangle ineq
For $f$ in $C^1[a,b]$, define $p(f)= \parallel f'\parallel _{\infty}$. Show that $p$ is non-negative, homogeneous, and satisfies the triangle inequality. Why is it not a norm? -I can easily show the ...