# Tagged Questions

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### Necessity of completeness of the inner product space in Riesz representation theorem

I wanted to find a counter example to show that the completeness of the inner product space is necessary in Riesz representation theorem. Please give an example of a bounded linear functional $T$ on ...
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### On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
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### Topology induced by Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
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### Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
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### Let $z_1 = x_1 +iy_1$ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $< z_1 , z_2> = x_1x_2+y_1y_2$

Let $z_1 = x_1 +iy_1$ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $\langle z_1 , z_2 \rangle = x_1x_2+y_1y_2$ For non zero $z_1$ and $z_2$ prove ...
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### Let V be an inner product space. If $x⊥y$, then show that

Let $V$ be an inner product space. If $x_i ⊥ x_j$ when $i\neq j$, then show that $$\Bigg\Vert\sum_{i=0}^n x_i\Bigg\Vert^2\ =\sum_{i=0}^n \Vert x_i \Vert^2.$$
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### Fredholm alternative and orthonormal basis

The following question relates to the Fredholm alternative: Let $K:H \rightarrow H$ be a compact linear operator and let $I$ be the identity operator. Notation: $N$ is the nullspace and $R$ is the ...
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### Show that if the Riesz map is surjective on $H$, then $H$ is a Hilbert space
Let $H$ be a vector space equipped with an inner product $(\cdot, \cdot)$ and $f:H\to H',\ f(x)=(\cdot,x)$ surjective. Now, why $H$ is a Hilbert space? The other direction is clear by Riesz' ...