Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven. General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, many Banach spaces lack a notion analogous to an orthonormal basis. That is, we require. In Banach spaces, a large part of the study involves the dual space : the space of all continuous linear maps from the space into its underlying field, so-called functionals.
A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation. This is explained in the dual space article. Also, the notion of derivative can be extended to arbitrary functions between Banach spaces.
Functional Analysis – Universität Innsbruck
The uniform boundedness principle or Banach—Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn—Banach theorem and the open mapping theorem , it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators and thus bounded operators whose domain is a Banach space , pointwise boundedness is equivalent to uniform boundedness in operator norm.
The theorem was first published in by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn. Theorem Uniform Boundedness Principle. Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y.
Annals of Functional Analysis
If for all x in X one has. There are many theorems known as the spectral theorem , but one in particular has many applications in functional analysis. Theorem:  Let A be a bounded self-adjoint operator on a Hilbert space H.
This is the beginning of the vast research area of functional analysis called operator theory ; see also the spectral measure. There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces.
The Hahn—Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting".
The open mapping theorem , also known as the Banach—Schauder theorem named after Stefan Banach and Juliusz Schauder , is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map. It therefore seems unlikely that Glu 74 and Arg 77 are involved in dimerization. As their mutation clearly affected function, it is more likely they have others roles to play.
Charged amino acids within the membrane can have many different functions including affecting function by snorkeling 22 , interacting with phospholipids 23 neutralizing helix dipoles 12 and modulating helix orientation While the role Glu 74 and Arg 77 are playing is not yet completely clear, and may await the elucidation of the structure of the protein, insights might be gained by comparison with other sequences and by examination of other proteins with known structures.
The sequence consists of Asp 59 and Arg 62 that are separated by Phe and Val. Examination of the Pa NhaP crystal structure suggests that Asp 59 and Arg 62 face the periplasmic side. As a consequence, they appear to induce a bend in TM 3 opening the periplasmic funnel of the protein.
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Arg 62 is within interacting distance of the C-terminal of the TM 12, which is one of the discontinuous helices of the metal transport domain. TM 3 belongs to the dimerization domain and TM 12 belongs to the metal transport domain. Therefore, Arg 59 might bridge an interaction between the two domains.
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It is therefore possible that Arg 77 mediates interactions with the transport domain however the structure of Sp NHE1 would need to be elucidated to confirm this hypothesis. Two other amino acids that we examined in this study were Arg and Arg Here we demonstrated that it is essential for activity with both Arg and Arg necessary for transport in S. Mutation of either residue to Ala or negatively charged Glu, abolished or greatly reduced the ability of these proteins to confer salt tolerance on yeast.
Mutational data, kinetics analysis and molecular dynamics simulation suggested that this basic residue Lys is essential for electrogenic transport of cations, and participates in cation binding in NapA 27 , However, another study has proposed an indirect involvement of the corresponding Lys of NhaA, providing both a structural and functional role, but not an exclusive requirement for electrogenic ion transfer It should be noted that in both cases the basic residue is important for efficient ion transport as well as the thermal stability.
We were curious as to why yeast species have a basic residue pair and other species do not. We note that except for the E. The salt bridge interaction of these Arg residues is in vicinity of the helical discontinuous region of these transmembrane segments.
It may be that in yeast, the number of Arg residues is finely tuned. Introduction of an additional Arg within transmembrane segments of Sp NHE1 can result in an inactive or severely defective protein 6. In species other then yeast, the arginines are present in two different helices. One Arg may provide the structural constraints and the other may provide a proper environment for metal transport.
We have earlier 31 suggested that binding of cations could be based on coordination of the cations by a crown ether-like cluster of polar amino acids as originally hypothesized by Boyer Previously 6 , Ala scanning mutagenesis showed that the individual residues Asp , Glu and Glu are not critical to protein function. Mutation number 9 NQQ neutralized the charges of all 3 residues to similar sized and uncharged amino acids but only had a very minor effect on function of the protein.
This resulted in a mutant protein that was not able to confer salt tolerance. This was likely not due to deletion of EQ itself, but was a cumulative effect of the replacements of Asp , Glu , Glu , and Glu Additionally, mutation EL alone did not have a large effect on the ability of the protein to confer salt tolerance. It thus seems that there is a requirement for negatively charged residues in this region, but each individual residue is not absolutely required.
Again, confocal images of NQQQ show that the mutation did not hinder the localization to the plasma membrane. The inactivity of the NQQQ mutant is thus due to compromised activity. The replacement of these amino acids had no affect on the ability of the protein to confer salt tolerance. These results are in agreement with the suggestion that acidic residues are required in this region, but an absolute position is not strictly required.
One or two acidic amino acids maintain protein function. Here, we returned one Glu in the replacement sequence and amino acid Asp was retained.
There was a slight effect on conveyance of salt tolerance with reducing the acidic residues to a single amino acid whereas when two acidic residues remained, with the replacement of the fragment with the SOS1-like Asp containing sequence, function was completely maintained. Overall, it seems that the region is not highly specific for the location of acidic charges, but requires one and ideally two acidic amino acids to maintain function.
We suggest that this negatively charged density due to the acidic residues, may be acting to aid in attraction of cations to the membrane protein pore. A cation or ion coordination sphere has been demonstrated in other membrane proteins, including the acetylcholine receptor 36 , 37 and NhaA Overall, our results have determined that EL6 plays an important role in the ability of the Sp NHE1 protein to confer salt tolerance in yeast. In the present study, we used our earlier predicted model of Sp NHE1 Future studies will determine if the crystal or cryo-electron microscope structure of Sp NHE1 corresponds with our predicted structure of the protein.
Restriction enzymes were obtained from New England Biolabs, Inc. The knock out strain was maintained on sodium deficient minimal KMA medium or yeast extract adenine YEA as described earlier 7 , 8. Wherever indicted LiCl or NaCl were added to media at the concentrations indicated. An NdeI site was removed by silent mutation to aid in cloning. Transformation of the plasmids with and without mutations, was by electroporation into the Sp NHE1 knock out strain sodura 4 A cell-DNA mixture was transferred to a cuvette for electroporation 0. Cells were harvested by centrifugation and spread onto KMA agar containing 1.
Transformed S. To examine growth on plates, serial dilutions of cells were inoculated onto agar with KMA medium containing leucine supplemented with either NaCl or LiCl at the indicated concentrations. Most mutations created or removed a restriction enzyme site A total of 13 types of mutations were done. To examine the cumulative effect of mutation of both to Ala, the primer set E74R77A was used 3.
Another mutation was done to exchange to position of the two amino acids, the primer set E74R77RE was used 4. To investigate the roles of Arg and Arg initially each were individually mutated to Ala RA and RA primers, 5, 6 and then later the charge was changed from positive to negative Glu with the primer set RE and RE 7, 8.
Asp, Glu, Glu, and Glu are on putative extracellular loop six discussed below. To examine their importance in the protein first we converted Asp , Glu , Glu , to their corresponding amide counterparts using the primer set DN,EQ,EQ. Additionally, to examine the role of Glu in further detail we made the mutation EL The modified sequence was chosen such that it contains at least one acidic residue as well as roughly the same amino acid length that of the deleted sequence.
Cell lysates were made from transformed yeast cultures grown as described above. Western blotting of nitrocellulose transfers used an anti-GFP polyclonal antibody a generous gift of Dr. Luc Berthiaume, Dept. Secondary antibody was goat anti-mouse antibody peroxidase-conjugated Bio-Can, Mississauga, Canada.
X-ray film detected protein reactivity and was via the Amersham enhanced chemiluminescence western blotting and detection system. The alignment was prepared using ESpript Confocal imaging was used to examine localization of S. Images were acquired using Volocity software Improvision Inc.